Trans. Amer. Math. Soc. arXiv:2005.05470v1 [math.SP] 11 May 2020 · 2020-05-13 ·...

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HIDDEN SYMMETRIES IN NON-SELF-ADJOINT GRAPHS

AMRU HUSSEIN

Abstract. On finite metric graphs all realizations of the Laplace operator in L2 defined by boundaryconditions are studied most of which are non-self-adjoint. In [Hussein, Krejcirık, Siegl, Trans. Amer.

Math. Soc., 367(4):2921–2957, 2015] a notion of regularity of boundary conditions by means of theCayley transform of the matrices parameterizing these has been proposed. The main point here isthat not only the existence of the Cayley transform is essential for basic spectral properties, but alsoits poles and its asymptotic behaviour which can be characterized using the quasi-Weierstrass normalform which exposes some “hidden” symmetries of the system. Thereby, generators of C0- and analyticsemigroups and C0-cosine operator function can be characterized. On star-shaped non-compact graphsa characterization of generators of bounded C0-groups and thus of operators similar to self-adjoint onesis obtained.

1. Introduction

Laplace operators on finite metric graphs are quasi-one-dimensional models used not only in quantummechanics, where they are called quantum graphs, see e.g. [BK13] and references therein, but also inthe modelling of a variety of wave and diffusion dynamics, compare e.g. [Mug14,AM94,Kuc02] just tomention a few. The question addressed here is how the boundary or matching conditions on the verticesinfluence spectral properties of such mostly non-self-adjoint Laplacians in particular with regard to thewell-posedness of the linear heat, wave and Schrodinger equations on finite metric graphs. In particular,for the L2-space on the graph all generators of C0-semigroups and of C0-cosine operator function arecharacterized in terms of boundary conditions. For generators of bounded C0-groups at least for stargraphs such a characterization is obtained.

Self-adjoint boundary conditions are well-studied, and these can be characterized by classical ex-tension theory. Considering for instance a star-shaped graph consisting of d copies of [0,∞) gluedtogether at the origins as illustrated in Figure 1 (b), then the deficiency indices are equal to d, andlinear boundary conditions of the form

Aψ(0) +Bψ′(0) = 0

define a self-adjoint Laplacian in the L2-space over the graph, i.e., a realization of minus the secondderivative operator on each edge, if and only if

AB∗ = BA∗ and Rank(AB) = d,

cf. [KS99,Har00]. Here general boundary conditions including all non-self-adjoint ones are treated, andthe condition Rank(AB) = d, i.e., that one has d linearly independent boundary conditions, is necessaryfor basic spectral properties such as a non-empty resolvent set. However this is not sufficient, and inaddition some kind of regularity assumption is needed. For self-adjoint boundary conditions defined bymatrices A,B the scattering matrix is given by minus the Cayley transform of these matrices

S(k,A,B) = −(A+ ikB)−1(A− ikB), k > 0,

and it appears in many instances such as in the Green’s function or in the eigenvalue equation for thecorresponding Laplacian, cf. e.g. [KS99,KS06]. This motivated the notion of regularity of boundaryconditions proposed in [HKS15] by Krejcirık, Siegl, and the author where boundary conditions definedby matrices A,B are regular if their Cayley transform exists for some k ∈ C. This regularity impliesmany basic spectral properties while irregular boundary conditions can lead to some extreme spectralproperties such as empty spectrum and empty resolvent sets, cf. [HKS15].

1

http://arxiv.org/abs/2005.05470v1

2 AMRU HUSSEIN

[0,∞)[0, a]

(a) General finite metricgraph

[0,∞). ..

(b) Star graph (c) Compact metricgraph

Figure 1. Finite metric graphs

The key observation communicated here is that for regular boundary conditions the quasi-Weierstrassnormal form reveals some additional structure or “hidden” symmetry of the operator, and thereby allowsfor a systematic analysis of Laplacians defined by regular boundary conditions. The quasi-Weierstrassnormal form has been introduced in the context of algebraic differential equations by Berger, Ilch-mann, and Trenn, see [BIT12], and for regular boundary conditions it implies that there are equivalentboundary conditions defined by

A = G−1

[

L 00 1Cd−m

]

G and B = G−1

[

1Cm 00 NB

]

G,

for a similarity transform G, a block decomposition Cd = Cd−m × Cm, a nilpotent matrix NB ∈C(d−m)×(d−m), and a matrix L ∈ Cm×m. Therefrom one can deduce that the Cayley transform isuniformly bounded outside its poles if and only if NB ≡ 0, and the order of the poles of S(·, A,B) isrelated to NB and the Jordan normal form of L, see Subsection 3.2.

Taking then advantage of this structure one can characterize the generators of C0-semigroups in L2-spaces. These are defined exactly by those regular boundary conditions with NB ≡ 0, see Theorem 4.1below which is one of the major findings of this article. Moreover, these C0-semigroups extend to analyticsemigroups, and its generators also generate C0-cosine operator function. Thus the question of the well-posedness of the heat and wave equation is completely answered in terms of boundary conditions. Aheuristic explanation for the effect of NB 6= 0 is discussed in Subsection 4.1.

There has been a number of results on some classes of boundary conditions defining generators ofC0-semigroups where in the form

〈−ψ′′, ψ〉 =ˆ

|ψ′|2 + 〈ψ′(0), ψ(0)〉Cd

the trace of the derivative ψ′(0) can be balanced by terms involving only the trace ψ(0), e.g. if ψ′(0) =−Lψ(0), see [KFMS07,Mug07,Mug14]. However, as it turns out, not all generators of C0-semigroupsfall under this category, and it seems that these cases have not been addressed in the literature so far.However, altering the scalar product such operators can still be associated with closed sectorial formsof Lions type. This is elaborated in Subsection 4.2. For irregular boundary conditions the resolvent haseven an exponential growth and for boundary conditions with NB 6= 0, the resolvent does not decayfast enough or even grows polynomially, see Subsections 4.4 and 4.5.

It should be highlighted that the characterization of generators of C0-semigroups involves – just asfor self-adjointness – only the boundary conditions and not any other geometric feature. In contrast, forthe question if a Laplacian is the generator of a bounded C0-group, i.e., if it is similar to a self-adjointoperator, the geometry matters. Here, it is shown, that for the relatively simple geometry of a star graph,where each edge is identified with [0,∞), similarity to a self-adjoint operator can be characterized interms of the poles of the Cayley transform, see Theorem 5.1 below. For other geometries this is nolonger true, and using the same boundary conditions in a different geometric setting can destroy thesimilarity relation, see Subsection 5.1. The result obtained here can be seen as a generalization tofinite star graphs of theorems on point interactions on the real line by Mostafazadeh, see [Mos06], and

HIDDEN SYMMETRIES IN NON-SELF-ADJOINT GRAPHS 3

Grod and Kuzhel [GK14]. The study of operators similar to self-adjoint ones is relevant in the so-callednon-Hermitian quantum mechanics, see e.g. [Ben07, SGH92,Mos10] and the references therein. SomePT -symmetric operator fall also in this class, see e.g. [AFK02,AK05,KS10,Kre19] and many more. Ageneralization to star graphs is proposed and analysed in [AKU15]. The general situation of non-self-adjoint extensions of symmetric operators with only absolutely continuous spectrum have been studiedin [Kis09,Kis11,KF00], see also the references therein.

In the subsequent Section 2 basic concepts and definitions are recapitulated. The notion of regularityof boundary conditions and properties of Cayley transforms are elaborated in Section 3. Some differencesto the classical Birkhoff-Tamarkin theory and its notion of regularity are exemplified in Subsection 3.4.The characterization of semigroup generators is discussed in Section 4. There also the completenessof the root vectors for compact graphs is addressed. Section 5 deals with the similarity to self-adjointoperators on star graphs.

2. Laplacians on finite metric graphs

2.1. Finite metric graphs. A graph is a 4-tuple

G = (V , I, E , ∂)

with the set of vertices V , the set of internal edges I, and the set of external edges E , and summingup E ∪ I is the set of edges. The boundary map ∂ assigns to each internal edge i ∈ I an ordered pairof vertices ∂(i) = (∂−(i), ∂+(i)) ∈ V × V , where ∂−(i) is called its initial vertex and ∂+(i) its terminalvertex. Each external edge e ∈ E is mapped by ∂ onto a single, its initial vertex. A graph is called finiteif |V|+ |I|+ |E| <∞.

A graph G can be endowed with the following metric structure. Each internal edge i ∈ I is associatedwith an interval [0, ai], with ai > 0, such that its initial vertex corresponds to 0 and its terminal vertexto ai. Each external edge e ∈ E is associated to the half line [0,∞) such that ∂(e) corresponds to 0.The numbers ai are called lengths of the internal edges i ∈ I and they are summed up into the vector

a = {ai}i∈I ∈ (0,∞)|I|.

The 2-tuple consisting of a finite graph endowed with a metric structure is called a metric graph (G, a).The metric on (G, a) is defined via minimal path lengths, and (G, a) is compact if E = ∅. The notationsused here largely parallel those in the works of Kostrykin and Schrader, see e.g. [KS06]. The constructionof a metric graph as quotient space is discussed by Mugnolo in [Mug19], where a metric graph consistsof intervals glued together at their end points according to the structure of a given graph.

2.2. Function spaces on finite metric graphs. A function ψ : (G, a) → C can be written as

ψ(x) = ψj(x), where ψj : Ij → C with Ij =

{

[0, aj], if j ∈ I,[0,∞), if j ∈ E ,

where with a slide abuse of notation ambiguities on the vertices are admitted. Occasionally, alsoψj(x) = ψj(xj) is written where x = xj ∈ Ij . Equipping each edge of the finite metric graph with theone-dimensional Lebesgue measure one obtains a measure space, and then one defines

ˆ

Gψ :=

∑

i∈I

ˆ ai

0

ψ(xi) dxi +∑

e∈E

ˆ ∞

0

ψ(xe) dxe,

where dxi and dxe refers to integration with respect to the Lebesgue measure on the intervals [0, ai] and[0,∞), respectively. Given a finite metric graph (G, a), one then considers the Hilbert space

L2(G, a) :=⊕

j∈I∪EL2(Ij ;C) with 〈ψ, ϕ〉 =

ˆ

Gψ ϕ.

4 AMRU HUSSEIN

Using the Sobolev spaces H1(Ij ;C) and H2(Ij ;C) on each edge, one defines

H1(G, a) :=⊕

j∈I∪EH1(Ij ;C) with 〈ψ, ϕ〉H1 = 〈ψ, ϕ〉 + 〈ψ′, ϕ′〉,

H2(G, a) :=⊕

j∈I∪EH2(Ij ;C) with 〈ψ, ϕ〉H2 = 〈ψ, ϕ〉H1 + 〈ψ′′, ϕ′′〉,

H20 (G, a) :=

⊕

j∈I∪EH2

0 (Ij ;C),

where ψ′ and ψ′′ denote the edgewise defined first and second distributional derivatives,

(ψ′)j (x) =ddxψj(x) and (ψ′′)j (x) =

d2

dx2ψj(x) for j ∈ E ∪ I, x ∈ Ij ,

and H20 (Ij ;C) with j ∈ E ∪ I denotes the set of all ψj ∈ H2(Ij ;C) with

ψj(0) = 0, ψ′j(0) = 0, for j ∈ E , ψj(0) = 0, ψ′

j(0) = 0, ψj(aj) = 0, ψ′j(aj) = 0, for j ∈ I.

The notations are sometimes shortened to L2(G), H1(G), H2(G), and H20 (G).

2.3. Laplacians and boundary conditions. One defines maximal and minimal Laplace operators inL2(G, a) by

∆maxψ = ψ′′ with Dom(∆max) = H2(G, a) and

∆minψ = ψ′′ with Dom(∆min) = H20 (G, a).

It is known that the operator ∆min is a closed symmetric operator with deficiency indices (d, d), where

(2.1) d := |E|+ 2|I|,and its Hilbert space adjoint is (∆min)∗ = ∆max; see e.g. [BEH08, Section 4.8]. The scope here aregeneral realizations ∆′ of the Laplacian on metric graphs with

(2.2) ∆min ⊂ ∆′ ⊂ ∆max.

These realizations ∆′ can be discussed in terms of boundary or matching conditions imposed at theendpoints of the edges. To this end, one defines the traces

ψ =

{ψe(0)}e∈E{ψi(0)}i∈I{ψi(ai)}i∈I

, ψ′ =

{ψ′e(0)}e∈E

{ψ′i(0)}i∈I

{−ψ′i(ai)}i∈I ,

and [ψ] :=

[

ψψ′

]

for ψ ∈ H2(G, a),

and one introduces the auxiliary Hilbert space

K ≡ K(E , I) = KE ⊕K−I ⊕K+

I

with KE = C|E| and K(±)I = C|I|. Then for ψ ∈ H2(G, a) one has ψ, ψ′ ∈ K, and [ψ] ∈ K2. Since for the

quotient space Dom(∆max)/Dom(∆min) ∼= K2, any realization ∆′ satisfying (2.2) is associated with asubspace M ⊂ K2 such that

∆′ = ∆(M), where ∆(M)ψ = ψ′′ and Dom(∆(M)) = {ψ ∈ H2(G, a) : [ψ] ∈ M}

3. Boundary conditions and Cayley transforms

3.1. Parametrization of boundary conditions. Recall that there are various ways to parametrisethe subspaces M ⊂ K2, and some have been summarized in [HKS15, Section 3]. The number of linearlyindependent boundary conditions imposed is equal to dimM. In particular if dimM ≥ d, for thedeficiency index d given in (2.1), then there exist A,B ∈ Hom(K) such that

M = M(A,B) := Ker(A, B) and ∆(A,B) := ∆(M(A,B)),

where

(A, B) : K2 → K, (A, B)(χ1, χ2)T = Aχ1 +Bχ2 for χ1, χ2 ∈ K.


An equivalent description of Dom(∆(A,B)) by the linear boundary conditions defined by A,B is then

(3.1) Dom(∆(A,B)) = {ψ ∈ H2(G, a) : Aψ +Bψ′ = 0}.Note that the parametrisation by the matrices A and B is not unique. Indeed, operators ∆(A,B)and ∆(A′, B′) agree if and only if the corresponding spaces M(A,B) and M(A′, B′) agree. Boundaryconditions defined by A,B and A′, B′ are called equivalent if M(A,B) = M(A′, B′). Notice thatthe boundary conditions are equivalent if there exists an invertible operator C ∈ GL(K) such thatsimultaneously

A′ = CA and B′ = CB.

3.2. Regular boundary conditions and the quasi-Weierstrass normal form. Operator pencilsA − λB with λ ∈ C appear in many contexts such as generalized eigenvalue problems or differentialalgebraic equations, see e.g. [BIT12] and the references therein. A basic regularity assumption is thefollowing.

Definition 3.1 (Regularity of operator pencils). Let A,B ∈ Cd×d, d ∈ N. The operator pencil A− λBis called regular if detA+ λB 6= 0 for some λ ∈ C, and irregular otherwise.

For matrices A,B defining a regular operator pencil, one can define the Cayley transform

S(k,A,B) := − (A+ ikB)−1 (A− ikB)

for all k ∈ C except an at most finite set, because det(A− ikB) is a non-vanishing polynomial of degreeat most d. In fact the Cayley transform has been the starting point for the introduction of a notion ofregular boundary conditions in [HKS15, Definition 3.2] without being aware that the analogous notionof regular operator pencils already had existed in a different context.

Definition 3.2 (Regularity of boundary condition, cf. Definition 3.2 in [HKS15]). Let A,B ∈ Hom(K)with Rank(A, B) = d. Then the boundary conditions defined by A,B are called regular if A − λB isregular, and irregular otherwise.

For self-adjoint Laplacians on star graphs with I = ∅, S(k,A,B) for k > 0 has the interpretation ofa scattering matrix, cf. e.g. [KS99]. Note that equivalent boundary conditions have the same Cayleytransform, and in turn boundary conditions can be recovered via

AS := −1

2(S(k,A,B)− 1) and BS :=

1

2ik(S(k,A,B) + 1) .

Remark 3.3. Regularity of the operator pencil A − λB already implies Rank(AB) = d, cf. [HKS15,Proposition 4.2] for the case I = ∅. The notion of regularity of boundary conditions assumes that thereare d linearly independent boundary conditions, i.e., Rank(AB) = dimM(A,B) = d. In fact, dimM <d leads to an under-determined while dimM > d leads to an over-determined system. This is reflected bythe spectrum since σ(∆(M)) = C for dimM 6= d, see [HKS15, Proposition 4.2]. Definition 3.2 requiresdimM = d for irregular boundary conditions, while the cases with dimM 6= d are out of scope here.However, having the d linearly independent boundary conditions, i.e., dimM = d, is not sufficientfor many spectral properties since irregular boundary conditions can lead to wild spectral features,see [HKS15, Section 3.4] and this is also reflected in Theorem 4.1 below characterizing C0-semigroupgenerators.

The quasi-Weierstrass normal form for regular operator pencils A−λB has been introduced in [BIT12]in the context of differential algebraic equations. Its proof is based on Wong sequences of subspaces.It is the key ingredient to the analysis presented here, and it reveals many properties of the Cayleytransform.

Proposition 3.4 (Quasi-Weierstrass normal form, cf. Theorem 2.6 in [BIT12]). Let A,B ∈ Cd×d and

let the operator pencil A− λB, λ ∈ C, be regular. Then there exists 0 ≤ m ≤ d and invertible matricesF,G such that

A = F

[

L 00 1Cd−m

]

G and B = F

[

1Cd 00 NB

]

G,

6 AMRU HUSSEIN

where NB ∈ C(d−m)×(d−m) is nilpotent and L ∈ Cm×m. These matrices are not uniquely determined.

Recall that for self-adjoint boundary conditions one has a unique parametrization, cf. [Kuc04, The-orem 6], by

A = L+ P, B = P⊥ = (1− P ) with P ∗ = P = P 2, P⊥L = LP⊥,(3.2)

and L = L∗. Hence, −∆(A,B) is associated with the closed symmetric form δP,L defined by

(3.3) δP,L[ψ] =

ˆ

G|ψ′|2 − 〈LP⊥ψ, P⊥ψ〉K, dom(δP,L) = {ϕ ∈ H1(G, a) | Pϕ = 0}.

For general possibly non-selfadjoint L with P⊥L = LP⊥ the form δP,L is sectorial, though in generalnon-symmetric, and it has been investigated in detail in the context of semigroups on networks, seee.g. [Mug14, Chapter 6 and 7] and the references therein, and [Hus14] where a characterization is givenwhen general non-self-adjoint boundary conditions are of this form. One observes that regular boundaryconditions in general are not of the form needed to define δP,L. However, at least if NB = 0, there areequivalent boundary conditions which are similar to such sectorial boundary conditions, i.e.,

A = G−1(L+ P )G, B = G−1P⊥G, with P ∗ = P = P 2, P⊥ = (1− P ), P⊥L = LP⊥.

This motivates the following definition, and it will turn out, see Theorem 4.1 below, that this notionsuggesting a relation to sectorial operators is indeed justified.

Definition 3.5 (Quasi-sectorial boundary conditions). Regular boundary conditions A,B are calledquasi-sectorial if NB = 0 in the quasi-Weierstrass normal form.

Note that the condition that NB = 0 includes the case of m = d, and m = d − 1 implies NB = 0.For a matrix M ∈ Cn×n and λ ∈ σ(M) denote by γM (λ) the maximal length of Jordan chains to theeigenvalue λ or equivalently the size of the largest Jordan block to the eigenvalue λ.

Lemma 3.6 (Poles of the Cayley transform). Let A,B ∈ Hom(K) define regular boundary conditions,where m, L and NB are given by the quasi-Weierstrass normal form. Then the function

S(·, A,B) : C → Hom(K), k 7→ S(k,A,B)

is meromorphic with at most max{m+1, d} different poles, and each pole falls under one of the followingcases with poles of order zero being removable singularities:

(a) If k ∈ σ(−iL) \ {0}, then k is a pole of order γL(k);(b) If 0 ∈ σ(−iL), then 0 is a singularity of order max{γL(0)− 1, γNB(0)− 1, 0};(c) If 0 /∈ σ(−iL), then 0 is a singularity of order max{γNB(0)− 1, 0}.

Proof. By the quasi-Weierstrass normal form one has for regular boundary conditions

S(k,A,B) = G−1

[

S(k, L,1) 00 S(k,1, NB)

]

G.(3.4)

Now, without loss of generality on can assume that L and NB have the Jordan normal form, where thenecessary similarity transforms can be incorporated into G−1 and G. Recall that for a nilpotent matrixN with Nn = 0 and Nn−1 6= 0, i.e., γN (0) = n, one has that

(λ1+N)−1 = 1λ

(

1+ N−λ + . . .+ Nn−1

(−λ)n−1

)

.

Hence if n ∈ N0 is such that NnB = 0 and Nn−1

B 6= 0, then

S(1, NB; k) = −(1+ ikNB)−1(1+ ikNB − 2ikNB)

= −1+ 2ik(1+ ikNB)−1NB

= −1+ 2ikNB + 2(ik)2(−1)N2B + . . .+ 2(ik)n−1(−1)n−2Nn−1 for k 6= 0.

(3.5)


If L = λ1+N for λ ∈ C with n ∈ N0 such that Nn = 0 and Nn−1 6= 0, then for λ 6= 0

S(L,1; k) = −((λ+ ik)1+N)−1((λ + ik)1+N − 2ik1)

= −1+ 2ik((λ+ ik)1+N)−1

= −1+2ik

λ+ ik

(

1+N

−(λ+ ik)+ . . .+

Nn−1

(−1)n−1(λ+ ik)n−1

)

for λ+ ik 6= 0,

(3.6)

and for λ = 0

S(L,1; k) = −1+ 2

(

1+N

−ik + . . .+Nn−1

(−1)n−1(ik)n−1

)

for ik 6= 0.(3.7)

If L = λ1 for λ ∈ C this simplifies to become

S(L,1; k) = − (λ− ik)

(λ+ ik)1, if λ 6= 0 and S(L,1; k) = 1, if λ = 0.

So, if k ∈ σ(iL) \ {0}, then each Jordan block JL,k of L to the eigenvalue k contributes a pole at kof order γJL,k

(0), and the Jordan blocks of L to different eigenvalues and also S(1, NB; k) do not havea pole at k which proves (a).

In case (b), the Jordan blocks JL,0 of L to 0 contribute poles of order γJk,0(0) − 1, and the Jordan

blocks of JNB ,0 contribute poles at 0 of order max{0, γNB(0) − 1}. In the case (c), only the Jordanblocks of JNB ,0 contribute poles at 0 of order max{0, γNB(0)− 1}.

Moreover, in (3.4) S(k, L,1) can have at most m different poles since L can have at most m pairwisedisjoint non-zero eigenvalues, and S(k,1, NB) for NB 6= 0 has only a pole a zero. So the set of polesconsists of at most m+ 1 points. Outside its poles the functions S(·, A,B) is holomorphic. �

Lemma 3.7 (Uniform boundedness of the Cayley transform). Let A,B define regular boundary condi-tions. Then the following statements are equivalent:

(a) The boundary conditions defined by A,B are quasi-sectorial.(b) For fixed ε > 0, the function

C \ {Bε(p1), . . . , Bε(pl)} → Hom(K), k 7→ S(k,A,B),

where p1, . . . , pl are the poles of S(·, A,B), is uniformly bounded.

Proof. By (3.4) it is sufficient to consider S(k, L,1) and S(k,1, NB). First,

S(k, L,1) = −( 1ikL+ 1)−1( 1

ikL− 1),

and hence using the Neumann series

‖(L+ ik1)−1(L− ik1)‖ ≤ 2

1− ‖L‖/k ≤ 4 for |k| ≥ 2‖L‖,

and due to continuity S(·, L,1) is bounded on B2‖L‖(0) \ {Bε(p1), . . . , Bε(p1)}.Second, if there exists a cyclic vector y with NBy 6= 0 and N2

By = 0 – which is equivalent to NB 6= 0– one has by (3.5) that S(k,1, NB)y = −y − 2ikNy, and hence for some c > 0

‖S(A,B; k)‖ ≥ c(1 + |k|) k ∈ C,

while for m− d > 0 and NB = 0 one has S(k,1, NB) = −1. �

Remark 3.8. In the case d = 1, i.e., |E| = 1 and |I| = 0, all boundary conditions with dimM = 1 areof the form

ψ(0) = 0 or α · ψ(0) + ψ′(0) = 0, α ∈ C.

and so all boundary conditions are regular and define operators associated with forms as in (3.3) of thetype δ1,0 and δα,1, respectively. Irregular and non-quasi-sectorial boundary conditions occur only ford ≥ 2. The case of an interval, i.e., d = 2 with E = ∅ is covered by the classical Birkhoff-Tamarkin theorywhich will be discussed below in Subsection 3.4, and the case d = 2 with I = ∅, i.e., a point interactionon the real line, is discussed in detail in [HM20], yet without the theory of the quasi-Weierstrass normalform at hand. Second order elliptic boundary value problems in domains in Rn for n ≥ 2 are often

8 AMRU HUSSEIN

reduced to conditions for the normal derivatives only, thus simplifying the problem to a one dimensionalone, where the more involved cases discussed here are not occurring.

3.3. Topology of boundary condition. The complex Grassmanian Gr(n,m) is the set of all n-dimensional subspaces of Cm. Using unitary groups, it can be identified with

Gr(n,m) = U(m)/(U(n)× U(m− n))

which induces also a differentiable structure on Gr(n,m) which makes it a compact manifold withdimGr(n,m) = n(m − n). Here, each boundary condition defined by M = M(A,B) can be identifiedwith a point [M] ∈ Gr(d, 2d). A metric on Gr(d, 2d) is defined by

d(M,M′) = ‖PM − PM′‖, M,M′ ∈ Gr(d, 2d),

where PM and PM′ are the orthogonal projections in K2 onto M and M respectively.

Remark 3.9. It is a classical result from extension theory that the set of self-adjoint boundary condi-tions is a submanifold of Gr(d, 2d) isomorphic to U(d) which is of real dimension d2 while the manifoldof all boundary conditions Gr(d, 2d) ∼= U(2d)/(U(d)× U(d)) has real dimension 2d2.

Parts of the following lemma are covered by [KPS08, Lemma 3.2]. It allows one to draw backconvergence in Gr(d, 2d) to convergence in Hom(K).

Lemma 3.10. One has dimM(A,B) = d if and only if AA∗+BB∗ is invertible. For dimM(A,B) = dthe orthogonal projection in K2 on M(A,B)∗ is given by

PM(A,B)⊥ =

[

A∗

B∗

]

(AA∗ +BB∗)−1[

A B]

.

In particular if dimM(An, Bn) = d and dimM(A,B) = d and An → A and Bn → B as n → ∞ inHom(K), then also PM(An,Bn) → PM(A,B).

Combing this with the fact that A,B define regular boundary conditions if and only if dim(M(A,B)) =d and KerA∩KerB 6= {0}, compare [HKS15, Proposition 3.3], one obtains the following characterizationof regular boundary conditions.

Corollary 3.11 (Characterization of regular boundary conditions). The boundary conditions definedby A,B ∈ Hom(K) are regular if and only if both

AA∗ +BB∗ and A∗A+B∗B

are invertible.

Proof of Lemma 3.10. Lemma 3.2 in [KPS08] deals with the necessary condition. For completeness, theproof is repeated here. First, note that

M(A,B)⊥ = Ker(A, B)⊥ = Ran

[

A∗

B∗

]

⊂ K2,

and since dimK2 = 2d, one has that dimM(A,B) = d if and only if M(A,B)⊥ = d, and henceKer(A∗, B∗)T = KerA∗ ∩ KerB∗ = {0}. In particular for dimM(A,B) = d the form defined byξ 7→ 〈B∗ξ, B∗ξ〉K+〈A∗ξ, A∗ξ〉K is coercive, and hence AA∗+BB∗ is invertible. It is now straight forwardto check, that RanPM(A,B)⊥ = M(A,B)⊥ and P 2

M(A,B)⊥ = PM(A,B)⊥ and PM(A,B)⊥ = P ∗M(A,B)⊥ . The

convergence follows directly from the formula for the projection since PM(A,B) = 1−PM(A,B)⊥ and thecontinuity of the composition and inversion operators. �

Remark 3.12. Let A,B define irregular boundary conditions, i.e., dim(M(A,B)) = d and KerA ∩KerB 6= {0}. Then the pencil A − λB is irregular, and it follows by taking adjoints that det(A∗ −λB∗) = 0 for all λ ∈ C, and hence also the pencil A∗ − λB∗ is irregular. However, by Lemma 3.10KerA∗ ∩KerB∗ = {0}. This is no contraction to the characterization of irregular boundary conditionssince the boundary conditions defined by A∗, B∗ violate the rank condition. More preciously, onehas dimM(A∗, B∗) = dim(Ran(A,B)T )⊥ > d since dim(KerA ∩ KerB) ≥ 1. Note that in general∆(A,B)∗ 6= ∆(A∗, B∗), compare [HKS15, Subsections 3.6 and 3.7].


Proposition 3.13. The set of regular boundary conditions is an open, connected and dense submanifoldof Gr(d, 2d). For d ≥ 2 the set of quasi-sectorial boundary conditions is a dense subset, and neitheropen nor closed.

Proof. By Corollary 3.11

Cd×2dreg : = {(A, B) ∈ C

d×2d : A,B define regular boundary conditions}= {(A, B) ∈ C

d×2d : | det(AA∗ +BB∗)|+ | det(A∗A+B∗B)| 6= 0},

and since (A, B) 7→ | det(AA∗ + BB∗)| + | det(A∗A + B∗B)| is continuous Cd×2dreg ⊂ Cd×2d is an open

submanifold. Now, consider the quotient map

q : Cd×2dd := {(A,B) ∈ C

2d×d : Rank(A, B) = d} → Gr(d, 2d), (A, B) 7→ [M(A,B)],

which identifies q(A,B) = q(A′, B′) if M(A,B) = M(A′, B′). By Lemma 3.10 this is continuous, andin fact the topology on Gr(d, 2d) coincides with the quotient topology induced by q. Hence the set of

regular boundary conditions q(Cd×2dreg ) ⊂ Gr(d, 2d) is open.

As in the proof [HKS15, Theorem 3.10], one can show that any boundary conditions A,B withdimM(A,B) = d can be approximated by

Aε := A and Bε := B + εPKerB, ε > 0,

where PKerB is the orthogonal projection in K to KerB. In particular Aε, Bε define quasi-sectorialboundary conditions which implies the density of quasi-sectorial boundary conditions.

To prove that the set of regular boundary conditions in Gr(d, 2d) is connected, consider for regularboundary conditions A,B the decomposition of A with respect to KerA and (KerA)⊥ and define

Aε :=

[

P(KerA)⊥AP(KerA)⊥ 0PKerAAP(KerA)⊥ εPKerA

]

and Bε := (1 − ε)B, ε ∈ [0, 1]

where A0 = A, B0 = B and Aε is invertible for all ε ∈ (0, 1], hence (Aε, Bε) is regular for all ε ∈ [0, 1].The following map

γ : [0, 1] → Cd×2dd , ε 7→ (Aε, Bε), hence also q ◦ γ : [0, 1] → Gr(d, 2d), ε 7→ M(Aε, Bε)

is continuous. Moreover, A1 is invertible and B1 = 0, so M(A1, B1) defines the regular Dirichletboundary conditions, and hence the manifold of regular boundary conditions is connected.

To show that the set of quasi-sectorial boundary conditions is not closed for d ≥ 2, consider forinstance

A = 1, B = N, and Aε = 1, Bε = N + ε1 for ε > 0.

Then for nilpotent N , the matrices A,B define regular but not quasi-sectorial boundary conditionswhile Aε, Bε define quasi-sectorial boundary conditions for ε > 0 because then Bε is invertible. HoweverM(Aε, Bε) → M(A,B) in Gr(d, 2d).

Conversely, to show that the set of non-quasi-sectorial boundary conditions in Gr(d, 2d) is not closedfor d ≥ 2, let

A = 1, B = 0, and Aε = 1, Bε = εN for ε > 0.

Then A,B define quasi-sectorial boundary while for nilpotent N the matrices Aε, Bε define regular butnon-quasi-sectorial boundary conditions for ε > 0 since then εN is nilpotent, butM(Aε, Bε) → M(A,B)in Gr(d, 2d). �

3.4. Comparison to the Birkhoff-Tamarkin theory. A theory for boundary value problems onintervals has been initiated at the beginning of the 20th century by Birkhoff in the works [Bir08b,Bir08] and then continued by Tamarkin and many others. It is elaborated in classical textbooks like[DS71, Chapter XIX] or [Nai67]. The Birkhoff-Tamarkin theory focuses on the convergence of theeigenfunction expansion for non-self-adjoint boundary conditions. This theory is still developing, see forinstance [Loc00] and [Fre12] for a survey on Birkhoff-irregular problems, and the respective referencestherein, and it also ramified including parameter dependent eigenvalue problems, see e.g. [MM03].

10 AMRU HUSSEIN

Central to the Birkhoff-Tamarkin theory is a notion of regularity of boundary conditions often referredto as Birkhoff-regularity which is related to the behaviour of determinants related to the eigenvalueequation. The Birkhoff-Tamarkin theory applies to general n-th order differential operators on intervals.Note that for the second derivative operator on an interval [0, a] this is different from the notion ofregularity promoted here, compare [HKS15, Section 3.3]. The difference can be illustrated also using[DS71, page 2344f.] where all Birkhoff-irregular boundary conditions are characterized to be of the form

ψ′(0) + γψ′(a) + αψ(0) + βψ(a) = 0 and ψ(0)− γψ(a) = 0, or

ψ′(a) + γψ′(0) + αψ(0) + βψ(a) = 0 and ψ(a)− γψ(0) = 0, α, β, γ ∈ C,

where the terms + . . . in [DS71, page 2344f.] are interpreted here as the terms with α and β. In matrixform this translates to

A =

[

1 −γα β

]

and B =

[

0 01 −γ

]

.

This is irregular in the sense of Definition 3.2 if KerA∩KerB 6= {0} which holds if α = c and β = −cγfor some c ∈ C, i.e, β = −αγ. If α, β 6= 0 then A is invertible, and therefore one has even quasi-sectorialboundary conditions. In particular, using Theorem 4.17 below, it follows that Birkhoff-regularity is notequivalent to the generation of C0-semigroups, but it gives only one inclusion. If α = β = γ = 0, thenthe boundary conditions are regular but not quasi-sectorial.

There is also a system version of the Birkhoff-Tamarkin theory which is presented for instance in thebook by Naimark [Nai67, Chapter III S 8.4]. To highlight the difference to the theory presented here,consider the second derivative operator on two intervals [0, 1] with boundary conditions

A =

1 0 0 00 1 0 00 0 1 −10 0 0 0

and B =

0 0 0 00 0 0 00 0 0 00 0 1 1

.

This corresponds to

ϕ1(0) = 0, ϕ2(0) = 0, and ϕ1(1) = ϕ2(1), ϕ′1(1) = −ϕ′

2(1),

and −∆(A,B) equivalent to the Dirichlet Laplacian on [0, 2].Translating this into the formalism of Naimark, one has in the notation of Naimark n = 2 (order of

the operator) and m = 2 (number of edges), and the boundary conditions are of the form

U1(ϕ) = U10(ϕ) + U11(ϕ), and U2(ϕ) = U20(ϕ) + U21(ϕ),

where for the orders of the boundary conditions k0 = 0 and k1 = 1 one has

U10(ϕ) = A1ϕ(0), U11(ϕ) = B1ϕ(1), and

U20(ϕ) = A2ϕ′(0) +A20ϕ(0), U21(ϕ) = B2ϕ

′(1) +B20ϕ(1)

with

A1 =

[

1 00 1

]

, B1 =

[

0 00 0

]

, A2 =

[

0 00 0

]

, A20 =

[

0 00 0

]

, B2 =

[

0 01 1

]

, B20 =

[

1 −10 0

]

.

Then regularity of the boundary conditions is defined with ω1 = i and ω2 = −i via

Φ(s) =

[

(A1 + sB1)ωk1

1 (A1 + (1/s)B1)ωk1

2

(A2 + sB12)ωk2

1 (A2 + (1/s)B2)ωk2

2

]

=

1 0 1 00 1 0 10 0 0 0is is −is −is

, s ∈ C,

where boundary conditions are regular if for

detΦ(s) = ϕ−2s−2 + ϕ−1s

−1 + ϕ0 + ϕ1s1 + ϕ2s

1

one has ϕ−2 6= 0 and ϕ2 6= 0 which is not satisfied here since detΦ(s) = 0.Herefrom, one sees that the Birkhoff-regularity can produce artefacts when inserting artificial edges.

This indicates that this notion of regularity is not always compatible with a geometric interpretation of a


system of ordinary differential equations such as graphs. When looking for generators of C0-semigroupsit leaves out many relevant cases. Moreover, while the Birkhoff-regularity is a powerful tool to check forRiesz basis properties of eigenfunctions, its verification is usually quite laborious. In contrast using forinstance Corollary 3.11, one can check for regularity in the sense of Definition 3.2 on the level of linearalgebra.

4. Semigroup generation

Using the notion of quasi-sectorial boundary conditions introduced in Definition 3.5, one obtains acomplete picture for which Laplacians the first and second order Cauchy problems

∂tψ −∆(A,B)ψ = 0, t > 0, ψ(0) = ψ0, and

∂2t ψ −∆(A,B)ψ = 0, t > 0, ψ(0) = ψ0, ψt(0) = ϕ0,

are well-posed in terms of C0-semigroups, analytic semigroups, and C0-cosine operator functions, seee.g. [ABHN01] for the definitions, and also [BC13] for a discussion of the differences between cosinefamilies and semigroups. Note that the same set of boundary conditions can be imposed on graphs withdifferent geometries as illustrated in Figure 4, where moreover, the length of the internal edges can bevaried. However, the characterization given below depends only on the boundary conditions and not onthe geometry.

Theorem 4.1 (Characterisation of generators of C0-semigroups). Let (G, a) be a finite metric graph.Then ∆(A,B) is the generator of a C0-semigroup in L2(G, a) if and only if A,B define quasi-sectorialboundary conditions. If ∆(A,B) is the generator of a C0-semigroup in L2(G, a), then this semigroupextends to an analytic semigroup, and ∆(A,B) generates a C0-cosine operator function.

Remarks 4.2. (a) Generators ofC0-cosine operator functions generate also C0-semigroups, see [ABHN01,Theorem 3.14.17 and its proofs]. Therefore, Theorem 4.1 characterizes also the generators of C0-cosine operator functions, and since analytic semigroups are in particular C0-semigroups, it charac-terizes also generators of analytic semigroups.

(b) Note that Theorem 4.1 deals with general C0-semigroups with

‖e∆(A,B)t‖ ≤ Ceµt, t ≥ 0, for some C > 0 and µ ∈ R.

The question when one can chose C = 1, i.e., when ∆(A,B) is quasi-dissipative (or equivalently−∆(A,B) is quasi-accretive) has been discussed in [Hus14, Theorem 3.1], and this is the case if andonly if there are equivalent boundary conditions of the form (3.2). Bounds on µ > 0 are discussedin [Hus13]. The semigroup is bounded if one can chose µ = 0, and the case µ = 0 and C = 1 hasbeen characterized in [Hus14, Theorem 3.2]. The boundedness of the semigroup has been addressedat least for point interactions in [HM20, Theorem 3.1 (d)].

There has been a number of results on some classes of boundary conditions defining generatorsof analytic semigroups, see e.g. [KFMS07, Mug07, Mug14, KPS08] which however do not include acharacterization. Some boundary conditions of the form (3.2) defining generators of C0-cosine operatorfunction are considered also in [Mug14], and also in [EKF19] using different methods and consideringmore general elliptic second order on graphs. Apart from this there is an extensive literature on the waveequation on networks, see e.g. [AM94,AM84,LLS94,Kuc02,DZ06,KFMS07,JMZ15,Klo12,KPS12b] andthe references therein, and even more so for the heat equations, see e.g. [vB88,Bob12,KPS08,KPS12,KPS07,BFN16] and the many the references therein.

4.1. Nilpotent matrices and resolvent estimates. Before discussing the proof of Theorem 4.1, itis instructive to have a closer look on the mechanism which prevents certain regular non-quasi-sectorialboundary conditions from defining a generator of a C0-semigroup. Consider the interval [0, 1] and theregular boundary conditions defined by

A =

[

1 00 1

]

and B =

[

0 0−1 0

]

,

12 AMRU HUSSEIN

Figure 2. Same boundary conditions for different geometries

i.e., ψ(0) = 0 and ψ(1)− ψ′(0) = 0. Then dimM(A,B) = 2 and det(A+ ikB) = 1 for all k ∈ C, i.e.,A,B are regular. Moreover,

S(k;A,B) = −[

1 02ik 1

]

, k ∈ C.

By Lemma 3.7 and Theorem 4.17 ∆(A,B) does not generate a C0-semigroup on L2(G, a). This isan illustrative example for regular but non-quasi-sectorial boundary conditions. It can be found in[Bir08, p.383] as well as in [DS71, Ex. XIX.6(d)]. There, it is an example of so-called intermediateboundary conditions. It stands here as an exemplification of boundary conditions of the type A = 1 andB = N where N is nilpotent.

The same boundary conditions defined by A,B given above can also be considered on a graph withonly two external edges, i.e., I = ∅ and |E| = 2 with E = e1 ∪ e2. Then the resolvent kernel is given by

rM(x, y; k) =i

2k

{[

eik|x1−y1| 00 eik|x2−y2|

]

+

[

eikx1 00 eikx2

]

S(k;A,B)

[

eiky1 00 eiky2

]}

.

Therefrom, the strategy of the proof of Theorem 4.1 becomes apparent when one takes into accountLemma 3.7, see also [HM20, Lemma 5.3]. In particular in the case considered here, for k = iκ withκ→ ∞, the resolvent does not exhibits the decay (−∆(A,B)+κ2)−1 . O(1/κ2) necessary for semigroupgenerators.

For a more heuristic interpretation one can consider the resolvent problem (−∆(A,B) − k2)ψ = fmore directly using the Dirichlet Laplacian −∆(1, 0) on [0, 1]. Then using the Ansatz

ψ = ψ1 + ψ0 where ψ1 = (−∆(1, 0)− k2)−1f, Im k > 0,

one finds that

ψ0(x) = sin(kx)ψ′1(0)

sin(k)− 1, Im k > 0,

and since−∆(1, 0)ψ1 = f+k2ψ1, where−∆(1, 0) = D∗0D0 withD0ψ = ψ′ and Dom(D0) = H1

0 ([0, 1];C),

ψ′1(x) = (D∗

0)−1(f + k2ψ1)(x) =

ˆ 1

x

f(y) + k2ψ1(y)dy.

where (D∗0)

−1 : L2([0, 1]) → RanD0. Using the trace or evaluation operator γ0 which maps ψ → ψ(0),this implies that

|ψ′1(0)| = |γ0 ◦ (D∗

0)−1(f + k2ψ1)| ≤ ‖γ0 ◦ (D∗

0)−1‖L(L2;C)

(

‖f‖L2 + ‖k2(−∆(1, 0)− k2)−1f‖L2.)

≤ C‖f‖L2

for some C > 0 and any k2 ∈ C \ Σ0,θ for θ ∈ (0, π/2), where Σ0,θ := {z ∈ C : | arg(z)| ≤ θ}. Here oneuses that the trace operator γ0 : H

1([0, 1];C) → C is bounded,

Ran(D∗0)

−1 = {ψ ∈ H1([0, 1];C) :

ˆ

[0,1]

ψ = 0},

and hence γ0 ◦ (D∗0)

−1 ∈ L(L2(G, a);C), and that the Dirichlet Laplacian generates an analytic semi-group. Summarizing, the trace ψ′

1(0) is given as a 0-th order operator applied to f the norm of whichcan be estimated independent of k.


More concretely, choosing for instance an eigenfunction of the Dirichlet Laplacian as right hand side

fκ(x) = sin(πκx), κ ∈ N, then ψ1(x;κ) =sin(πκx)

π2κ2 − k2, and ψ′

1(0;κ) =πl

π2κ2 − k2,

if k2 6= κ2π2. Then for k = iκ

‖ψ1(·;κ)‖ = O(1/κ2) while ‖ψ0(·;κ)‖ = |ψ′1(0;κ)|O(1/

√κ) = O(1/κ3/2) as κ→ ∞.

The role of the Dirichlet problem and the behaviour of the trace of the derivative of its solutionbecome even more apparent when A,B given above are considered on a graph with only two externaledges, i.e., I = ∅ and |E| = 2 with E = e1 ∪ e2. Then the resolvent problem can be rewritten to acoupled system

(− d2

dx21

− k2)ψ1 = f1, ψ1(0) = 0,

(− d2

dx21

− k2)ψ2 = f2, ψ2(0) = ψ′1(0).

This can be solved iteratively, to obtain first, using the Dirichlet Laplacian −∆(1, 0) on [0,∞), that

ψ1 = (−∆(1, 0)− k2)−1f1 if Im k > 0.

The second equation is then a boundary value problem with inhomogeneous Dirichlet boundary condi-tions a solution of which is given by

ψ2(x) = eikxψ′1(0) + ((−∆(1, 0)− k2)−1f2)(x), x ∈ e2, if Im k > 0,

and ψ′1(0) = γ0 ◦ ∂x1

(−∆(1, 0)− k2)−1f1, hence |ψ′1(0)| ≤ O(‖f1‖). Now for k = iκ with κ > 0

‖e−κxψ′1(0)‖L2 . O(‖f2‖L2/κ) as κ→ ∞.

This resembles the resolvent estimate of a first order operator and not that of a second order operator.Interchanging A and B, i.e., one can consider

A =

[

0 0−1 0

]

and B =

[

1 00 1

]

,

and then the resolvent behaviour is rather different, Since B is invertible, this defines quasi-sectorialboundary conditions. The resolvent problem can be rewritten to become

(− d2

dx21

− k2)ψ1 = f1, ψ′1(0) = 0,

(− d2

dx21

− k2)ψ2 = f2, ψ′2(0) = ψ1(0),

which can be solved iteratively, where now using the Neumann Laplacian −∆1(0, 1) on [0,∞)

ψ1 = (−∆1(0, 1)− k2)−1f1 for Im k > 0.

The second equation is then a boundary value problem with inhomogeneous boundary conditions asolution of which is given by

ψ2 =eikx

ikψ1(0) + (−∆2(0, 1)− k2)−1f2 for Im k > 0,

and ψ1(0) = γ0(−∆1(0, 1)− k2)−1f1. For k = iκ with κ > 0 it follows that

|ψ1(0)| = |γ0(−∆1(0, 1) + 1)−1/2(−∆1(0, 1) + 1)1/2(−∆1(0, 1) + κ2)−1f1| . O(‖f1‖/κ), κ→ ∞,

where one uses that the operator γ0(−∆1(0, 1)+1)−1/2 is bounded, and that by interpolation ‖(−∆1(0, 1)+1)1/2(−∆1(0, 1)+κ2)−1‖ . O(1/κ) as κ→ ∞. Hence ‖e−κxψ′

1(0)‖L2 . O(‖f2‖L2/κ2) as κ→ ∞ whichis compatible with the properties of C0-semigroup generators.

14 AMRU HUSSEIN

4.2. Quadratic forms for quasi-sectorial boundary conditions. In many instances spectral prop-erties of operator can be drawn back to quadratic forms. Here, integration by parts gives

〈−∆(A,B)ψ, ψ〉 =ˆ

G|ψ′|2 + 〈ψ′, ψ〉K, ψ ∈ Dom(∆(A,B)).

If the boundary conditions are of the form (3.2), then 〈ψ′, ψ〉K = 〈−LP⊥ψ, P⊥ψ〉K for ψ ∈ Dom(∆(A,B)),and hence this defines a sectorial form associated with −∆(A,B). However if this is not the case, thennot all the derivative terms at the vertices cancel out, and therefore the numerical range is not confinedto a sector, see for instance [Hus14]. This can be illustrated in the following example.

Example 4.3 (PT -symmetric point interaction). Let G be a graph consisting of two external edgesE = {e1, e2} and one vertex ∂(e1) = ∂(e2). Consider the boundary conditions defined by

Aτ =

[

1 −eiτ0 0

]

and Bτ =

[

0 01 e−iτ

]

for τ ∈ [0, π/2].

Identifying the graph with the real line and the vertex with zero, the boundary conditions correspond toψ(0+) = eiτψ(0−) and ψ′(0+) = e−iτψ′(0−). This example is included in the study of PT -symmetricpoint interactions in [AFK02] and was further investigated in [AK05] and [Sie09,Sie11].

The quadratic form defined by the operator −∆(Aτ , Bτ ) simplifies by integrating by parts and insertingthe boundary conditions to become

〈−∆(Aτ , Bτ )ψ, ψ〉 =ˆ

G|ψ′|2 + (1− e2iτ )ψ2(0)ψ′

2(0) ψ ∈ Dom(−∆(Aτ , Bτ )).

In particular, the derivative term cannot be avoided, and the numerical range is entire C for all τ ∈(0, π/2]. However, despite the numerical range for τ ∈ [0, π/2) the operator −∆(Aτ , Bτ ) is similar tothe self-adjoint Laplacian −∆(A0, B0) and hence the generator of an analytic semigroup. Note thatAτ , Bτ define regular boundary conditions with k-independent Cayley transform

S(Aτ , Bτ , k) =1

cos(τ)

[

i sin(τ) 11 −i sin(τ)

]

for τ ∈ [0, π/2).

Hence, this is an example of quasi-sectorial boundary conditions which are not in the form (3.2).

This exemplifies also the well-known fact that the numerical range of an operator is not stable undersimilarity transforms. It underlines that one cannot use always the usual L2-scalar product to relatesuch operators to forms.

The similarity transform for star graphs introduced in [HKS15, Section 6] gives for the case I = ∅ withquasi-sectorial boundary conditions a similarity transform to a Laplacian with boundary conditions ofthe form (3.2). The strategy here for the case I 6= ∅ is to take this similarity transform as an inspirationfor a localization procedure to construct an adjusted scalar product, and thereby to associate quasi-sectorial Laplacians with closed forms in this new setting.

To this end, consider first cut-off functions

χi+, χ

i−, χ

i0 : [0, ai] → [0, 1] with (χi

+)2 + (χi

−)2 + (χi

0)2 ≡ 1 for i ∈ I, and

χi−, χ

i0 : [0,∞) → [0, 1] with (χi

−)2 + (χi

0)2 ≡ 1 for i ∈ E ,

such that for amin := mini∈I ai if I 6= ∅ and amin > 0 arbitrary if I = ∅, one has

suppχi+ ⊂ [ai − amin/2, ai], suppχi

− ⊂ [0, amin/2], suppχi0 ⊂ [amin/4, ai − amin/4] for i ∈ I, and

suppχi− ⊂ [0, amin/2], suppχi

0 ⊂ [amin/4,∞) for i ∈ E .Define the auxiliary space

Haux := L2(G−)⊕ L2(G+)⊕ L2(GE )⊕ L2(G), L2(GC) :=⊕

j∈c(C)L2(Ij ;C), c(C) =

{

I, C ∈ {+,−},E , C = E .


Next, define some auxiliary operators. First, for χi+, χ

i−, χ

i0 as above let

χ : L2(G, a) → Haux, ψ 7→ χψ, χψ :=

{χiEψi}i∈E

{χi−ψi}i∈I

{χi+ψi}i∈I

{χi0ψi}i∈I∪E

,

where the properties of the cut-off functions translates to χ∗ ◦ χ = 1L2(G,a). Second, one definesidentification operators for i, j ∈ E ∪ I by

Iij : L2(Ij ;C) → L2(Ii;C), (Iijψj)(x) =

{

ψj(x), x ∈ Ii ∩ Ij ,0, x ∈ Ii \ Ij ,

that is, functions on the j-th edge are restricted or extended by zero and interpreted as functions onthe i-th edge. Note that in general IliIij 6= Ilj . Putting these identification operators together one candefine for R,S ∈ {E ,−,+} operators

IR,S = {Iij}i∈c(R),j∈c(S) : L2(GR) → L2(GS).

For G ∈ Cd×d invertible for d given by (5.3), one has a block structure G = (GR,S)R,S∈{E,−,+}, where

GR,S =∈ C|R|×|S| with C ∈ {R,S} and |C| = |I| for C ∈ {−,+} and |C| = |E| for C = E . Then one sets

MG : Haux → Haux, MGψ =

GE,EIE,E GE,−IE,− GE,+IE,+ 0G−,EI−,E G−−I−− G−+I−+ 0G+,EI+,E G+−I+− G++I++ 0

0 0 0 1

{ψi}i∈E{ψi}i∈I−

{ψi}i∈I+

{ψi}i∈I∪E

.

With the change of orientation operator on each internal edge defined by (I+ψ)i(x) = ψi(ai − x) fori ∈ I, one defines

I, J : Haux → Haux, I =

1 0 0 00 1 0 00 0 I+ 00 0 0 1

, J =

1 0 0 00 1 0 00 0 −1 00 0 0 1

,

which satisfy I∗ = I, I2 = 1, J∗ = J , J2 = 1, and by the chain rule

(Iψ)′ = JI(ψ′) for ψ ∈ H1(G−)⊕H1(G+)⊕H1(GE )⊕H1(G).With these preparations at hand, one considers the sequilinear form defined by

〈ψ, ϕ〉G,χ := 〈IMGIχψ, IMGIχϕ〉Haux , ψ, ϕ ∈ L2(G, a).

Lemma 4.4. Under the above assumptions 〈·, ·〉G,χ defines a scalar product in L2(G, a), and the inducednorm is equivalent to the norm induced by the canonical scalar product 〈·, ·〉.Proof. Linearity follows from the linearity of the scalar product in Haux along with the anti-linearity

〈ψ, ϕ〉G,χ = 〈ψ, χ∗IM∗GMGIχϕ〉Haux

= 〈χ∗IM∗GMGIχϕ, ψ〉 = 〈ϕ, ψ〉G,χ, ψ, ϕ ∈ L2(G, a).

To show the equivalence of norms observe first that

〈ψ, ψ〉G,χ = 〈χ∗IM∗GMGIχψ, ψ〉L2(G) ≤ ‖χ∗IM∗

GMGIχ‖‖ψ‖2L2(G) for ψ ∈ L2(G, a).Then, note that on the range of Iχ one has IliIij = Ilj , because supp(Iχψ)i ⊂ [0, amin /2] for all i ∈ I±∪Eand hence MHMGIχψ =MHGIχψ. In particular for G invertible MG−1MGIχψ = Iχψ. So, using thatI∗ = I, I2 = 1, and χ∗ ◦ χ = 1L2(G,a) one obtains for ψ ∈ L2(G, a)

〈ψ, ψ〉G,χ = 〈MGIχψ,MGIχψ〉Haux ≥ ‖MG−1‖−1〈Iχψ, Iχψ〉Haux = ‖MG−1‖−1〈ψ, ψ〉L2(G).�

Remark 4.5. The relation between the standard scalar product 〈·, ·〉 and 〈·, ·〉G,χ is given by

〈ψ, ϕ〉G,χ = 〈ψ,Θϕ〉 where Θ = χ∗IM∗GMGIχ

and Θ is self-adjoint and by Lemma 4.4 positive, and it plays the role of the metric operator.

16 AMRU HUSSEIN

Now one can relate the operator −∆(A,B) for quasi-sectorial boundary conditions A,B to a sesquilin-ear form with respect to the scalar product 〈·, ·〉G,χ. Consider

〈−ψ′′, ϕ〉G,χ = 〈−ψ′′, χ∗IM∗GMGIχϕ〉L2(G)

= 〈ψ′, (χ∗IM∗GMGIχϕ)

′〉L2(G) + 〈ψ′, χ∗IM∗GMGIχϕ〉K.

Recall that MHMGIχψ =MHGIχψ, in particular M∗GMGIχψ =MG∗GIχ. So,

(χ∗IMG∗GIχϕ)′ = (χ∗)′IMG∗GIχϕ+ χ∗(IMG∗GIχϕ)

′

= (χ′)∗IMG∗GIχϕ+ χ∗JIMG∗GJI(χϕ)′

= (χ′)∗IMG∗GIχϕ+ χ∗JIMG∗GIJχ′ϕ+ χ∗JIMG∗GIJχϕ

′

and with H := G∗G

(χ∗IMG∗GIχϕ)(x) ={χjE(xj)H

jiE,EIE,Eχ

iEψi(xi)}i,j∈E + {χj

E(xj)HjiE,−IE,−χ

i−ψi(xi)}i∈I,j∈E

+{χjE(xj)H

jiE,+IE,+χ

i+ψi(ai − xi)}i∈I,j∈E + {χj

−(xj)Hji−,EI−,Eχ

iEψi(xi)}i∈E,j∈I

+{χj−(xj)H

ji−−I−,−χ

i−ψi(xi)}i,j∈I + {χj

−(xj)HjiE,+IE,+χ

i+ψi(ai − xi)}i,j∈I

+{χj+(xj)H

ji+,EI+,Eχ

iEψi(ai − xi)}i∈E,j∈I + {χj

+(xj)Hji+−I+,−χ

i−ψi(ai − xi)}i,j∈I

+{χj+(xj)H

ji++I+,+χ

i+ψi(ai − xi)}i,j∈I + {χj

0(xj)χi0ψi(xi)}i,j∈E∪I .

Hence

χ∗IMG∗GIχϕ = G∗Gϕ and 〈ψ′, χ∗IMG∗GIχϕ〉K = 〈Gψ′, Gϕ〉K.The goal of the whole construction is to eliminate the trace term ψ′ which appears after integration byparts. Now, recalling that by the quasi-Weierstrass normal form

Dom(A,B) = {ψ ∈ H2(G) : (L+ P )Gψ + P⊥Gψ′ = 0},where L, P satisfy (3.2), one obtains for dom(δA,B) := {ψ ∈ H1(G) : P⊥Gψ = 0} that

〈Gψ′, Gϕ〉K = −〈LGψ,Gϕ〉K, ψ ∈ Dom(∆(A,B)), ϕ ∈ dom(δA,B),

and one defines the sesquilinear form for ψ, ϕ ∈ dom(δA,B) by

(4.1) δA,B[ψ, ϕ] := 〈MGIJχψ′,MGIJχϕ

′〉+ 〈MGIχ′ψ′,MGIχϕ〉+ 〈MGJIχψ

′,MGIJχ′ϕ〉

− 〈LGψ,Gϕ〉K.Recall that a quadratic form δ in a Hilbert space H with domain V is of Lions type if there is constantC > 0 such that | Im δ[ψ]| ≤ C‖ψ‖H‖ψ‖V for all ψ ∈ V , cf. [Mug14, Section 6.2] .

Lemma 4.6. Let A,B be quasi-sectorial boundary conditions, then the form δA,B is a closed densely de-fined sectorial form of Lions type in L2(G, a) with 〈·, ·〉G,χ as scalar product, and −∆(A,B) is associatedwith δA,B.

Proof. Since ⊕i∈I∪IC∞0 (Ij ;C) ⊂ dom(δA,B), and ⊕i∈I∪IC∞

0 (Ij ;C) ⊂ L2(G, a) is dense, also δA,B isdensely defined.

Next, one considers the leading term 〈MGIJχψ′,MGIJχψ

′〉, and one observes that

Im〈MGIJχψ′,MGIJχψ

′〉 = 0,

〈MGIJχψ′,MGIJχψ

′〉 ≤ ‖χ∗JIMG∗MGIJχ‖‖ψ′‖2,〈MGIJχψ

′,MGIJχψ′〉 ≥ ‖MG−1‖−1〈IJχψ′, IJχψ′〉 = ‖MG−1‖−1‖ψ′‖2,

where the two estimates are analogous to the ones in the proof of Lemma 4.4. Hence this term definesa densely defined closed symmetric form on dom δA,B. The other terms will be interpreted as relativelybounded perturbations.

For the second and third term one has

| Im〈MGIχ′ψ′,MGIχϕ〉| ≤ |〈MGIχ

′ψ′,MGIχϕ〉| ≤ ‖(χ′)∗IJMG∗GIJχ‖‖ψ′‖‖ϕ‖, and

| Im〈MGJIχψ′,MGIJχ

′ϕ〉| ≤ |〈MGJIχψ′,MGIJχ

′ϕ〉| ≤ ‖χ∗IJMG∗GIJχ′‖‖ψ′‖‖ϕ‖.


The trace term can be estimated by Agmon’s inequality in R, i.e., for ψi = u+ iv for i ∈ I, then

|ψi(0)|2 = u2(0) + v2(0) = 2

ˆ 0

ai

(χ−u)(x)(uχ−)′(x) + (χ−v)(x)(vχ−)

′(x)dx

≤ 2‖χu‖‖(uχ−)′‖+ 2‖χv‖‖(vχ−)

′‖= 2‖χu‖‖χ′

−u+ χu‖+ 2‖χv‖‖χ′−v + χv‖

≤ C(‖u‖+ ‖u′‖) + C(‖v‖+ ‖v′‖) ≤ C‖ψi‖‖ψ′i‖.

Analogous estimates hold for ψi(ai) using χ+ and for ψi(0) when i ∈ E using χE . Hence,

| Im〈LGψ,Gψ〉K| ≤ |〈LGψ,Gψ〉K| ≤ ‖G∗LG‖‖ψ‖2 ≤ C‖G∗LG‖‖ψ‖‖ψ′‖.Hence, the last three terms in (4.1) define relatively bounded perturbations of the form defined by

the first term. Therefrom, using the boundedness of the trace operator ψ 7→ ψ, and the completeness of

H1(G), one concludes that δA,B defines a densely defined closed form on dom(δA,B), cf. [Kat95, TheoremVI.3.4]. By the estimates on the imaginary part it follows that it is of Lions type. Moreover, the operator−∆(A,B) is associated with δA,B by the calculation above which give that

〈−∆(A,B)ψ, ϕ〉G,χ = δA,B[ψ, ϕ] for ψ ∈ Dom(∆(A,B)) and ϕ ∈ dom(δA,B).

In particular the assumptions of the first representation theorem [Kat95, Theorem VI.2.1] are satisfied,where one verifies that Dom(∆(A,B)) is a core of δ(A,B). �

4.3. Spectral enclosure. The location of the spectrum is a necessary condition for an operator to bea generator of a C0-semigroup.

Proposition 4.7 (Location of the spectrum). (a) If A,B define regular boundary conditions, then forany C > 0 there exists a c > 0 such that

σ(−∆(A,B)) ⊂ {z ∈ C : | Im z|+ c ≤ C(Re z)},i.e., it is contained in a sector;

(b) If A,B define quasi-sectorial boundary conditions, then

σ(−∆(A,B)) ⊂ {z ∈ C : c(Im z)2 − C ≤ Re z} for some C, c > 0,

i.e., the it is contained in a parabola around a positive half-axis.

Remark 4.8. In the context of non-self-adjoint operators, the location of the spectrum is not assignificant as in the self-adjoint case, and instead the emphasis is laid on pseudospectra which arerelated level sets of the norm of the resolvent. The notion of ε-spectra or pseudospectra goes back toLandau, see [Lan75], and since it has been investigated intensively, see e.g [KSTV15] and the referencestherein.

Proof of Proposition 4.3. In the case I = ∅ one has σ(−∆(A,B)) = σess(−∆(A,B)) ∪ σp(−∆(A,B)),where σess(−∆(A,B)) = [0,∞), see [HKS15, Proposition 4.11], σp(−∆(A,B)) is a finite set since theeigenvalue equation reduces to det(A+ ikB) for the root Im k > 0, while the residual spectrum is empty,see [HKS15, Proposition 4.6]. This already proves the claim for I = ∅.

Consider now the case I 6= ∅. By [HKS15, Proposition 4.7], k2 ∈ C \ [0,∞) is in the resolvent set,whenever for k with Im k > 0 the matrix A± ikB is invertible, and

det(1−S(k;A,B)T (k, a)) 6= 0, where T (k, a) =

0 0 00 0 eika

0 eika 0

.(4.2)

Note that

‖T (k, a)‖ ≤ e−(Im k)amin for k ∈ CIm>0, where amin := mini∈I

ai.

By Lemma 3.7 in case (a) and (b)

‖S(k,A,B)T (k, a)‖ ≤ |p(k)|e−(Im k)amin and ‖S(k,A,B)T (k, a)‖ ≤ Ce−(Im k)amin ,

respectively for k ∈ CIm>0 \ {Bε(p1), . . . , Bε(pl)}, where C > 0 is a constant and p is polynomial in k .

18 AMRU HUSSEIN

In case (b) for k ∈ CIm>0 with Im k > ln(C)/amin, there are no zeros of the secular equation (4.2).Hence all roots of eigenvalues are located in a strip parallel to the real axis. For k = reiϕ one has

Im k2 = r2 sin(2ϕ) = 2r sin(ϕ)r cos(ϕ) and Re k2 = r2 cos(2ϕ) = r2 cos2(ϕ)− r2 sin2(ϕ),

and hence

{k ∈ CIm>0 : Im k > C}2 = {z ∈ C : (Im z)2/C2 − C2 > Re z.}which is a parabola with vertex at −C2, see Figure 3 (a) and (c).

In case (a), one has |p(k)| ≤ α + β|k|d for some α, β ≥ 0 if p is a polynomial of degree d ∈ N0, andassuming C|Re k| < Im k for given C ∈ (0, 1), one has

‖S(k;A,B)T (k, a)‖ ≤ (α+ β√

C2 + 1 Im k)e−(Imk)amin .

Then there exists a c > 0 such that

‖S(k;A,B)T (k, a)‖ ≤ (α+ β√

C2 + 1 Im k)e−(Im k)amin ≤ 1

2for Im k ≥ c,

and hence by (4.2) for any such k one has that k2 ∈ ρ(−∆(A,B)). So,

{k ∈ CIm>0 : C|Re k| ≤ Im k, Im k > c}2 ⊂{z ∈ C : (Im z)2/C2 − C2 > Re z} ∩ {z ∈ C : Im z > |1/(1− C2)Re z|}.

That is the spectrum is contained in a sector as depicted in see Figure 3 (b) and (d). �

Re k

Im kCIm>0

(a) Im k ≤ C

Re k

Im kCIm>0

(b) Im k ≤ max{C|Re k|, c}

Rek2

Im k2

C

(c) Im k ≤ C

Re k2

Im k2

C

(d) Im k ≤ max{C|Re k|, c}

Figure 3. Gray indicates where roots of eigenvalues and eigenvalues can be located, respectively

Remark 4.9 (Erratum to Lemma 4.3 in [HKS15]). To show that the resolvent set for regular boundaryconditions is non-empty in [HKS15, Lemma 4.3], the author has assumed implicitly that S(iκ, A,B) isat least polynomially bounded as κ→ ∞. To justify this, one needs in fact Lemma 3.7 as shown in theproof of Proposition 4.7.

4.4. Resolvent estimates for regular boundary condition. The resolvent for regular boundaryconditions is an integral operator with kernel

rM(x, y; k) = r0(x, y; k) + r1M(x, y; k)

with {r0(x, y; k)}j,j′ = δj,j′i2k e

ik|xj−yj| and

r1M(x, y; k) =i

2kΦ(x, k, a) [1−S(k,A,B)T (k; a)]−1

S(k,A,B)Φ(y, k, a)T ,


where the Φ(x, k, a) is given by

Φ(x, k, a) :=

[

φE(x, k) 0 00 φ+(x, k) φ−(x, k)

]

,

respectively, with diagonal matrices

φE (x, k) = diag{eikxj}j∈E , φ+(x, k) = diag{eikxj}j∈I , and φ−(x, k) = diag{eik(aj−xj)}j∈I ,

and Φ(x, k)T denotes the transposed of Φ(x, k), see [KPS08, Lemma 3.10] and [HKS15, Proposition 4.7].

Lemma 4.10 (Upper bound on the resolvent for regular boundary conditions). Let A,B define regularboundary conditions, then for c, C are as in Proposition 4.7 (a) there exists a constant C′ > 0 such that

‖(−∆(A,B)− k2)−1‖ ≤ C′|k|d−2 for all k2 ∈ {z ∈ C : c(Im z)2 − C ≤ Re z}.Proof. Since

‖(−∆(A,B)− λ)−1‖ = ‖(−∆(A,B)− λ)−1 − (−∆(1, 0)− λ)−1‖+ ‖(−∆(1, 0)− λ)−1‖it is sufficient to estimate the operator norm of the integral operator defined by r1M(·, ·; k). From theproof of Proposition 4.7 (a) one deduces that

‖(1−S(k,A,B)T (k, a))−1‖ ≤ 2 for all k ∈ C \ {z ∈ C : c(Im z)2 − C ≤ Re z},and from the proof of Lemma 3.7 one obtains that for k away from the poles of S(k,A,B)

‖S(k,A,B)‖ ≤ C′′|k|d−1

for some C′′ > 0 Hence, for some C′ > 0

‖ˆ

Gr1M(x, y; k)f(y)dy‖ ≤ 1

|k|C′|k|d−1

Im k‖f‖ ≤ C′|k|d−2‖f‖, k ∈ {k ∈ CIm>0 : Im k > C}.

�

Lemma 4.11 (Lower bound on the resolvent for non-quasi-sectorial boundary conditions). Let A,Bdefine regular but not quasi-sectorial boundary conditions, then there exists a constant C > 0

‖(−∆(A,B) + κ2)−1‖ ≥ C as κ→ ∞.

Proof. Note first that since the Laplacian on L2(R) generates a contraction semigroup

‖ˆ

Gr0(x, y; iκ)ψ(y)dy‖ ≤ 1

κ2‖ψ‖ for κ > 0.

Then consider r1M(·, ·; iκ) together with Lemma 3.7. For

‖ˆ

Gr1M(x, y; iκ)ψ(y)dy‖ = sup

06=ϕ∈L2(G,a)

|〈´

G r1M(x, y; iκ)ψ(y), ϕ〉dy|

‖ϕ‖ ,

and since Φ(x, iκ, a) has only real entries for κ > 0

〈ˆ

Gr1M(x, y; iκ)ψ(y), ϕ〉dy =

1

2κ〈[1−S(k,A,B)T (k; a)]

−1S(k,A,B)

ˆ

GΦ(y, k, a)Tψ(y)dy,

ˆ

GΦ(x, k, a)Tψ(x)dx〉K.

Moreover since by the proof of Lemma 3.7 one has ‖S(iκ, A,B)‖ ≤ C|κ|d−1 as κ→ ∞, and ‖T (iκ; a)‖ =e−κamin, for any q ∈ (0, 1) there exists cq > 0 such that ‖S(iκ, A,B)T (iκ; a)‖ < q for κ ≥ cq, and henceusing Neumann series

[1−S(k,A,B)T (k; a)]−1

= 1+

∞∑

n=1

[S(k,A,B)T (k; a)]n.

20 AMRU HUSSEIN

Inserting this into the above gives

〈ˆ

Gr1M(x, y; iκ)ψ(y), ϕ〉dy

=1

2κ〈[1−S(k,A,B)T (k; a)]

−1S(k,A,B)

ˆ


ˆ

GΦ(x, k, a)Tϕ(x)dx〉K

=1

2κ〈S(k,A,B)

ˆ


ˆ

GΦ(x, k, a)Tϕ(x)dx〉K

+1

2κ

∞∑

n=1

〈[S(iκ, A,B)T (iκ; a)]n S(iκ, A,B)

ˆ


ˆ

GΦ(x, k, a)Tϕ(x)dx〉K.

There exists c′q > 0 such that

‖[S(iκ, A,B)T (iκ; a)]n S(iκ, A,B)‖ ≤ qn for κ ≥ c′q,

and hence for κ ≥ c′q

1

2κ|

∞∑

n=1

〈[S(iκ, A,B)T (iκ; a)]nS(iκ, A,B)

ˆ


ˆ

GΦ(x, k, a)Tψ(x)dx〉K|

≤ q

2κ(1− q)‖Φ(·, k, a)‖2‖ψ‖‖ϕ‖ → 0 as κ→ ∞,

where one uses that

‖φE(·, iκ)‖2 =1

2κ, ‖φ+(·, iκ)‖2 =

1

2κ(1− e−2κamin), ‖φ+(·, iκ)‖2 =

1

2κ(1− e−2κamin).

So, one can now focus on the leading term

1

2κ〈S(iκ, A,B)

ˆ

GΦ(y, iκ, a)Tψ(y)dy,

ˆ

GΦ(x, iκ, a)Tϕ(x)dx〉K.

Consider for αE = {αEj }j∈E , α− = {α−

j }j∈I , α+ = {α+j }j∈I

α =

αE

α−

α+

and ψα(x) =

[ {αEj e

−κxj}j∈E{α−

j e−κxj}j∈I + {α+

j e−κ(aj−xj)}j∈I

]

.

Then ψα ∈ L2(G, a),

‖ψα‖2 =1

2κ‖(1− e−2κaj)αE

j ‖2 +1

2κ‖(1− e−2κaj )α+

j ‖2 +1

2κ‖(1− e−2κaj )α−

j ‖2

+∑

j∈Iaje

−κaj2Reα+j α

−j ,

and

ˆ

GΦ(y, iκ, a)Tψα(y)dy = H(iκ, a)α, H(iκ, a) =

12κ1 0 00 1

2κ (1− e−2κa) a0 a 1

2κ (1− e−2κa)

.

Now, for NB 6= 0 from the quasi-Weierstrass normal form one chooses for v ∈ K with ‖v‖ = 1 andNBv 6= 0

α = H(iκ, a)−1G−1NBv and β = H(iκ, a)−1G∗NBv,

then

1

2κ〈S(iκ, A,B)

ˆ

GΦ(y, iκ, a)Tψα(y)dy,

ˆ

GΦ(x, iκ, a)Tψβ(x)dx〉K =

1

2κ〈v + 2κNBv,NBv〉K,

and since min{1/2κ, amin} . ‖H(iκ, a)‖ . max{1/2κ, amax} as κ→ ∞√2κmax{1/2κ, amax}−1 . ‖ψβ‖−1, ‖ψα‖−1 .

√2κmin{1/2κ, amin}−1‖G−1NBv‖−1.


Normalizing ϕα = ψα/‖ψα‖ and ϕβ = ψβ/‖ψβ‖, one obtains as for κ ≥ 1/2amax max{1/2κ, amax} =

amax that there is a constant C > 0 such that

1

2κ|〈S(k,A,B)

ˆ

GΦ(y, k, a)Tϕα(y)dy,

ˆ

GΦ(x, k, a)Tϕβ(x)dx〉K| ≥ C‖NBv‖2 as κ→ ∞,

while the other terms discussed above go to zero. �

4.5. Resolvent estimates for irregular boundary condition. Set as before amin := mini∈I ai ifI 6= ∅ and amin > 0 arbitrary if I = ∅.Lemma 4.12 (Lower bound on the resolvent for irregular boundary conditions). Let A,B be irregularboundary conditions, then

|eImkamin/2 − 3amin|amin|k2|

. ‖(−∆(A,B)− k2)−1‖ for Im k > 0,

where one uses the convention ‖(−∆(A,B)− k2)−1‖ = ∞ if k2 ∈ σ(−∆(A,B)).

The following example is an essential ingredient for the proof of Lemma 4.12.

Example 4.13. Consider the interval [0, a] and the irregular boundary conditions defined by

A =

[

1 00 0

]

and B =

[

0 01 0

]

.

Then dimM(A,B) = 2 = d and the boundary conditions correspond to

ψ(0) = 0 and ψ′(0) = 0.

This example is discussed in [DS71, Sec. XIX.6(b)] as totally degenerate boundary conditions. Notethat

−∆(A,B) = −D20, where D0ψ = ψ′ with Dom(D0) = {ψ ∈ H1([0, a]) : ψ(0) = 0},

and σ(D0) = ∅. Hence for any k ∈ C

(−∆(A,B) − k2)−1 = [−(D0 − ik)(D0 + ik)]−1 = −(D0 + ik)−1(D0 − ik)−1.

The resolvent for D0 is given by the well-known variation of constants formula

(D0 ± ik)−1ψ(x) =

ˆ x

0

e∓ik(x−y)ψ(y)dy, k ∈ C.

Inserting for instance ψ ≡ 1, one obtains

(−∆(A,B)− k2)−1ψ =1

k2(cos(kx)− 1) , k ∈ C \ {0}.

Proof of Lemma 4.12. Note that for I = ∅ one has ρ(−∆(A,B)) = ∅, see [HKS15, Section 3.4], andhence the claim follows.

So let I 6= ∅. Then, the first observation is that the function ψa defined by

ψa(x; k) =1

k2(cos(k(a− x))− 1)χ[0,a], ψa(·; k) ∈ H2([0,∞)) for k ∈ C \ {0}, a > 0,

because ψa(a; k) = ψ′a(a; k) = 0. Now, let v ∈ (KerA∩KerB)\{0} with v = [{vEj }j∈E , {v−j }j∈I , {v+j }j∈I ]

and ‖v‖ = 1, and then consider the function ψv ∈ H2(G) defined by

ψv(xj) =

{

vEj ψamin/2(x; k), j ∈ E ,

v−j ψamin /2(xj ; k) + v+j ψamin /2

(aj − xj ; k), j ∈ I.This satisfies

ϕv = ψa(0; k) · v and ϕv′ = ψ′

a(0; k) · v,

22 AMRU HUSSEIN

and hence ϕv ∈ Dom(∆(A,B)), and moreover it solves

(−∆(A,B)− k2)ψv = ϕv for ϕv(xj) =

{

vEj χ[0,amin/2], j ∈ E ,v−j χ[0,amin/2] + v+j χ[aj−amin/2],aj

, j ∈ I.

So, if k2 ∈ σ(−∆(A,B)), then one sets ‖(−∆(A,B) − k2)−1‖ = ∞ and if k2 ∈ ρ(−∆(A,B)), thenψv = (−∆(A,B) − k2)−1ϕv, and

‖ϕv‖ = (amin/2)‖v‖ and ‖v‖ |eImkamin − 3amin|

2|k2| . ‖ψv‖.

Hence the claim follows as in Example 4.13. �

4.6. Proof of Theorem 4.1. By the classifications of boundary conditions one can discuss now eachpossible case to show that only quasi-sectorial boundary conditions lead to the desired properties.

First, if dimM 6= d, then

σ(−∆(M)) = C

according to [HKS15, Proposition 4.2], and hence ∆(M) cannot be a generator. For A,B withdimM(A,B) = d defining irregular boundary conditions or regular boundary conditions which arenot quasi-sectorial by Lemma 4.12 and Lemma 4.11, respectively, one has

‖(−∆(A,B) + κ2)−1‖ ≥ C > 0 as κ→ ∞,

and therefore it cannot be a generator. Now, if A,B define quasi-sectorial boundary conditions, then byLemma 4.6, ∆(A,B) is the unique operator associated with the densely defined, closed, sectorial formδA,B. Hence it generate an analytic semigroup, cf. e.g. [Mug14, Theorem 6.15], in (L2(G, a), 〈·, ·〉G,χ)and by the equivalence of norms, see Lemma 4.4, it generates an analytic semigroup in L2(G, a) withthe standard scalar product, and in particular a C0-semigroup. Since the form is of Lions type it followsthat −∆(A,B) is the the generator of an C0-cosine operator function, cf. [Mug14, Theorem 6.18], andby equivalence of norms this carries over to L2(G, a). �

Combining Proposition 4.7 with Theorem 4.1 one obtains

Corollary 4.14. If −∆(A,B) is similar to a normal operator, then A,B define quasi-sectorial boundaryconditions.

Proof. Assume that −∆(A,B) is similar to a normal operator, then there is an equivalent norm ‖·‖nsuch that

‖(−∆(A,B)− k2)−1‖n = dist(k2, σ(−∆(A,B))), k2 ∈ ρ(−∆(A,B)).

since σ(−∆(A,B)) by Proposition 4.7 is contained in a sector, this implies that −∆(A,B) – takinginto account equivalence of norms – generates an analytic semigroup in L2(G, a) which by Theorem 4.1implies that −∆(A,B) is quasi-sectorial. �

Remark 4.15 (Normal Laplacians are already self-adjoint). A normal Laplacians in L2(G, a) satisfies

∆(A,B)∗∆(A,B) = ∆(A,B)∆(A,B)∗,

where both operators are self-adjoint bi-Laplacians. By the extension theory for self-adjoint bi-Laplacianswhich is analogous to the one for Lapalcians, see e.g. [GM20] and the references therein, it follows thatthis holds if and only if Dom(∆(A,B)∗) = Dom(∆(A,B)), i.e., the operator is already self-adjoint. Thisseems to be a general feature for extensions of closed symmetric operators.

Remark 4.16. To prove the generation of analytic semigroups and C0-cosine operator function, insteadof using the form δA,B constructed in Subsection 4.2, one can use also the explicit formula for the Green’sfunction given in Subsection 4.4 to prove suitable resolvent estimates.


4.7. Spectra on compact graphs. If G is compact, i.e., E = ∅, then σ(−∆(A,B)) is discrete becauseof the compact embedding Dom(−∆(A,B)) → L2(G, a). For an unbounded non-self-adjoint operator itis in general not self-evident that the resolvent set or the spectrum is non-empty. For irregular boundaryconditions, there both cases can occur, compare e.g. [HKS15, Example 3.2 and Example 3.1], and forregular boundary conditions one has by Proposition 4.7 that the resolvent set is non-empty.

The spectral projection on γ for a compact set γ ⊂ σ(−∆(A,B)) is defined by

EA,B(γ) =1

2πi

ˆ

Γ

(−∆(A,B)− z)−1dz,(4.3)

where Γ is a closed Jordan curve constructed such that it is positively oriented, and it bounds a finitedomain containing every point of γ and no point of σ(−∆(A,B))\γ. The range of the finite dimensionalrange of EA,B(λi) consists of the span of the eigenvectors and generalized eigenvectors to λi.

Proposition 4.17. Let I = ∅, i.e., (G, a) is compact. If the boundary conditions defined by A,B areregular, then the set of generalized eigenvectors of −∆(A,B) is complete in L2(G, a).Corollary 4.18. Let A,B be regular boundary conditions, then σ(−∆(A,B)) 6= ∅, and σp(−∆(A,B))is an at most countable infinite set, and for E = ∅, it is a countable infinite set.

Proof. The non-zero eigenvalues correspond to the zeros of the non-constant holomorphic functiondet(Z(k;A,B, a)), see [HKS15, Section 4.1] and Lemma 4.7, and hence the set of eigenvalues is atmost countable infinite. For E = ∅, by Theorem 4.17 (a) the set of generalized eigenvalues is complete.Since L2(G, a) is infinite dimensional, it follows that the dimension of the space spanned by generalizedeigenvectors is infinite. As the resolvent is compact, the multiplicity of each eigenvalue is finite and hencealso the set of eigenvalues. Since for E 6= ∅ one has that [0,∞) ⊂ σ(−∆(A,B)), see [HKS15, Proposition4.11], it follows in particular that the spectrum is non-empty. �

Combing this corollary with the results from [HKS15, Section 3 and 4] one can summarize the basicspectral properties of non-self-adjoint Laplacians on graphs in Table 1.

I = ∅ E = ∅ I 6= ∅, E 6= ∅σr(−∆(A,B)) = ∅ σr(−∆(A,B)) = ∅ σr(−∆(A,B)) ⊂ [0,∞)

or σr(−∆(A,B)) = ∅σe(−∆(A,B)) = [0,∞) σe(−∆(A,B)) = ∅ σe(−∆(A,B)) = [0,∞)

σp(−∆(A,B)) ⊂ C \ [0,∞) σ(−∆(A,B)) discrete σp(−∆(A,B)) discreteσp(−∆(A,B)) finite countable infinite at most countable infiniteand finite multiplicity eigenvalues eigenvalues

Table 1. Summary of spectral properties for regular boundary conditions

Example 4.19. Consider the interval [0, 1], and the regular, non-quasi-sectorial boundary conditions

A =

[

1 00 1

]

and B =

[

0 0−1 0

]

,

discussed in Subsection 4.1. The spectrum of −∆(A,B) consists only of eigenvalues of geometric mul-tiplicity one, where each eigenvalue is a solution of

sin(k) = k, k ∈ C,

with eigenfunction sin(kx) for k 6= 0 and x for k = 0, and a direct computation shows that thereare no cyclic vectors. One can ask if the set of eigenvectors forms even a Riesz basis. According to[DS71, Ex. XIX.6(d)] the eigenvalues are located asymptotically at the points

an2 + ibn lnn+ . . . , n ∈ N,

where the ratio of a and b is real. The question when on an interval functions of the form eikxk∈C form aRiesz basis for some set C ⊂ C has been characterized by Pavlov in [Pav79], and one of the conditions

24 AMRU HUSSEIN

is that | Im k| ≤ c for some c > 0 and all k ∈ C. So, the eigenvectors to this problem are complete butdo not form a Riesz basis.

Example 4.20 (Laplacian with nilpotent part). One can ask if for regular boundary conditions cyclicvectors can occur at all. To construct a Laplacian with nilpotent part let N ∈ C

n×n be a nilpotent matrixwith

Nn = 0 and Nk 6= 0 for k = 1, . . . , n− 1.

Consider the “pumpkin graph” with n edges each of length a illustrated in Figure 4. Then one can definequasi-sectorial boundary conditions in the block form

A =

[

−N 00 N

]

and B =

[

1 00 1

]

.

For v ∈ Cn one can define the function

ϕ(x) = eNxv, where ∂mx ϕ = Nmϕ, m ∈ N0.

Then

ϕ =

[

v{eNavi}i∈I

]

and ϕ′ =

[

Nv−N{eNavi}i∈I

]

.

Hence Aϕ+Bϕ′ = 0 and similarly A∂mx ϕ+B∂mx ϕ′ = 0. So,

∆(A,B)mϕ ∈ D(∆(A,B)) for all m ∈ N0, and ∆(A,B)[k/2]ϕ = N2[k/2]ϕ = 0.

For n ≥ 3, ∆(A,B)ϕ = N2ϕ 6= 0, so ϕ defines a cyclic vector to the eigenvalue 0.

Figure 4. “Pumpkin graph”

Proof of Proposition 4.17. There is a family of results on the completeness of the set of root vectorsof Hilbert-Schmidt operators, see for instance [DS71, Corollary IX.31] or [Loc00, Theorem 2.6.2] basedon the decay or at least polynomial growth of the resolvent. Here, the version of [Agr94, Lemma 2] isapplied to −∆(A,B) in the separable Hilbert space L2(G, a).

For E = ∅, the resolvent (−∆(A,B)−k2)−1 is compact for k2 ∈ ρ(−∆(A,B)), and the singular valuesof (−∆(A,B)− k2)−1 are the eigenvalues

{sj(k2;A,B)}j∈N with s1(k2;A,B) ≥ s2(k

2;A,B) ≥ . . . ≥ 0 counting multiplicties

of the non-negative self-adjoint compact operator

|(−∆(A,B) − k2)−1| = (((−∆(A,B) − k2)−1)∗∆(A,B))1/2 for k2 ∈ ρ(−∆(A,B)).

Lemma 4.21. If E = ∅, and A,B define regular boundary conditions, then (−∆(A,B) − k2)−1 is aHilbert-Schmidt operator for any k2 ∈ ρ(−∆(A,B)) 6= ∅, and

sj(k2;A,B) = O(1/j2) as j → ∞.

Proof. This follows already from the fact that the integral kernel of the resolvent, see 4.4, is continuousand for E = ∅ the graph is compact.

Note that for self-adjoint boundary conditions Asa, Bsa one can apply the classical Dirichlet-Neumannbracketing method, compare e.g. [BE09, Proposition 4.2], to obtain that the eigenvalues of−∆(Asa, Bsa)behave as O(j2) as j → ∞, and hence the singular values for any k ∈ ρ(−∆(Asa, Bsa))behave O(1/j2)


as j → ∞. If A,B defining regular boundary conditions, it follows by Proposition 4.7 that for anyself-adjoint boundary conditions ρ(−∆(A,B)) ∩ ρ(−∆(Asa, Bsa)) 6= ∅. For k2 ∈ ρ(−∆(A,B)) ∩ρ(−∆(Asa, Bsa)) the resolvent formula from Subsection 4.4 applies, and

(−∆(A,B) − k2)−1 − (−∆(Asa, Bsa)− k2)−1ψ =

ˆ

G(r1A,B(·, y; k)− r1Asa,Bsa

(·, y; k))ψ(y)dy

is a finite rank operator with dimension of its range smaller equal to d sinceˆ

Gr1A,B(·, y; k)ψ(y)dy ∈ span{ψ ∈ L2(G, a) : ψj = α+

j eikλxj + α−

j eikλxj , α±

j ∈ C}.

Therefore, compare e.g. [GK69, Corollary II.2.1], also

sj−d(k2;Asa, Bsa) ≤ sj(k

2;A,B) ≤ sj+d(k2;Asa, Bsa) for j ≥ d

and hence sj(k2;A,B) = O(1/j2) as j → ∞, and by the first resolvent identity this follows for all

k2 ∈ ρ(−∆(A,B)). �

So, by Lemma 4.21

lim infj→∞

sj(k2;A,B)j2 > 0

Hence, one needs 5 rays γ1, . . . , γ5 with angles between adjacent lines being smaller than π/2 such thatthere exists an N ≥ −1 such that

‖(−∆(A,B)− λ)−1‖ = O(N) as λ→ ∞ along each ray γj , j = 1, . . . , 5.

These can be chosen by Lemma 4.11 and Proposition 4.7 as illustrated in Figure 4.7. Hence by [Agr94,Lemma 2] the system of root vectors of −∆(A,B) is complete. �

Re k2

Im k2

C

Figure 5. Five rays with angles smaller π/2

5. Similarity to self-adjoint operators

The Schrdinger equation

∂tψ + i∆(A,B)ψ = 0, t > 0, ψ(0) = ψ0,

is well-posed if −∆(A,B) is the generator of a bounded C0-group or equivalently if −∆(A,B) is similarto a self-adjoint operator. For δ-potentials on the real line such characterizations have been pioneeredby the work of Mostafazadeh [Mos06], and further results for point interactions have been obtainedin [GK14] and [KZ17]. These results – obtained by a variety of methods such as Krein spaces andclassical criteria for quasi-self-adjointness from [Nab84,Mal85, vC83] – point into a similar directionwhere similarity depends on the spectrum of L in the parametrization 3.2, and the orders of somespectral singularities. In this sense the following characterization is a generalization of Mostafazadeh’scriterion, where the admissible range of spectra of L is illustrated in Figure 6. A particular classrotational invariant boundary conditions on star graphs has been studied in [AKU15].

26 AMRU HUSSEIN

Theorem 5.1 (Similarity on star graphs). Let G be a star graph with only external edges, i.e., I = ∅.Then −∆(A,B) is similar to a self-adjoint operator if and only if A,B define quasi-sectorial boundaryconditions, for the matrix L in the quasi-Weierstrass normal form

σ(L) ⊂ {z ∈ C : Re z < 0} ∪ [0,∞),

and for each l ∈ σ(L) ∩ [0,∞) the algebraic and geometric multiplicity agree.

Im k

Cσ(L)

Re k

Figure 6. Location of σ(L) for quasi-sectorial Laplacians similar to self-adjoint oper-ators in gray and dark gray where conditions on the multiplicity are required

Example 5.2 (Complex δ-interaction). Consider a graph with I = ∅ and |E| ≥ 2. Assume that theboundary conditions are defined by

A =

1 −1 0 · · · 0 00 1 −1 · · · 0 00 0 1 · · · 0 0...

......

......

0 0 0 · · · 1 −1−γ 0 0 · · · 0 0

and B =

0 0 0 · · · 0 00 0 0 · · · 0 00 0 0 · · · 0 0...

......

......

0 0 0 · · · 0 01 1 1 · · · 1 1

, γ ∈ C.

One can represent the boundary conditions by equivalent boundary conditions of the form 3.2 with P =1−P⊥, where P⊥ is the rank one projector onto (KerB)⊥, and L = − γ

|E|P⊥, cf. [Kuc04, Section 3.2.1]

for the case of real γ. Theorem 5.1 translates the result from [Mos06], see also [GK14, Example III],for the case |E| = 2 to |E| = d, that is, −∆(A,B) is similar to a self-adjoint operator if and only if

γ ∈ {z ∈ C : Re z > 0} ∪ (−∞, 0].

Example 5.3 (Complex δ′-interactions). As in example 5.2 consider a graph with I = ∅ and |E| ≥ 2,and now interchanging the boundary conditions, one considers

A =

0 0 0 · · · 0 00 0 0 · · · 0 00 0 0 · · · 0 0...

......

......

0 0 0 · · · 0 01 1 1 · · · 1 1

and B =

1 −1 0 · · · 0 00 1 −1 · · · 0 00 0 1 · · · 0 0...

......

......

0 0 0 · · · 1 −1−γ 0 0 · · · 0 0

, γ ∈ C,

compare [Kuc04, Section 3.2.3] for the case of real γ, and there are further δ′-type boundary conditionsdiscussed for instance in [Man10, Man15]. Equivalent boundary conditions are defined by A = P⊥

δ

and B = Lδ + Pδ, where Pδ = 1 − P⊥δ with P⊥

δ being the rank one projector from Example 5.2, andLδ = − γ

|E|P⊥δ . Hence, equivalent boundary conditions are

Aδ′ = −|E|γP⊥δ and Bδ′ = 1,


and therefore Lδ′ = − |E|γ P

⊥δ . Theorem 5.1 states that −∆(A,B) is similar to a self-adjoint operator if

and only if

γ ∈ {z ∈ C : Re z > 0} ∪ (−∞, 0],

because γ is in this region in C if and only if 1/γ lies there.In [GK14] the couplings on the real line with

1

2

[

a bc d

] [

ψ(0+) + ψ(0−)−ψ′(0+)− ψ′(0−)

]

=

[

ψ′(0+)− ψ′(0−)ψ(0+)− ψ(0−)

]

, a, b, c, d ∈ C,(5.1)

are considered, see [GK14, Lemma 2.1]. Identify R with a graph with I = ∅ and E = {e1, e2} withe1 taking the role of [0,∞) and e2 the one of (−∞, 0] (where both e1, e2 are identified with [0,∞)).Then ψ(0+) = ψ1(0), ψ(0−) = ψ2(0), and ψ

′(0+) = ψ′1(0), ψ

′(0−) = −ψ′2(0), and the coupling (5.1)

translates to

A =

[

a2

a2

c2 − 1 c

2 + 1

]

and B =

[

− b2 − 1 b

2 − 1− d

2d2

]

.

In particular the case of δ′-type conditions for a = b = c = 0 and d 6= 0 translates to

A =

[

0 0−1 1

]

, B =

[−1 −1− d

2d2

]

and equivalently A′ =1

d

[

1 −1−1 1

]

, B′ =

[

1 00 1

]

which is A′ =√2d P

⊥δ and B′ = 1. This is similar to a self-adjoint operator if and only if

d ∈ {z ∈ C : Re z < 0} ∪ [0,∞)

which is consistent with [GK14, Example IV], and the case d = 0 is included since it defines the self-adjoint Laplacian on R.

5.1. Geometry of a graph and similarity transforms. Semigroup generation, compare Theo-rem 4.1, such as self-adjointness, see e.g. [KS99], can be verified locally considering boundary conditionsat each vertex independent of the geometry. However, similarity to self-adjointness involves not onlythe local boundary conditions but also and foremost the geometry, and the symmetry of the graphplayed an essential role in the proof of Theorem 5.1. The following examples illustrate that similarityto self-adjoint operators can be caused by an interplay of the graph’s geometry and the local boundaryconditions, and the similarity can be prohibited as well by breaking such symmetries. In short for thegeneration of bounded C0-groups the geometry matters!

Example 5.4 (Broken symmetry). Consider the metric graph consisting of one internal edge of lengtha and one external edge. Impose the following boundary conditions defined by

Aτ =

1 −eiτ 00 0 00 0 1

and Bτ =

0 0 01 e−iτ 00 0 0

for τ ∈ [0, π/2).

In this case the Cayley transform is similar to a unitary matrix. However, this similarity does not carryover to similarity of operators. The eigenvalue equation becomes

ike−ika cos(τ) − ikeikai sin(τ) = 0 for k ∈ {z ∈ C : Im z > 0 or z ∈ (0,∞)}.For τ ∈ (0, π/2) and k = x+ iy this is equivalent to

i tan(τ) = e2ika = [cos(2ax)− i sin(2ax)]e2ya,

and therefore k2 is an eigenvalue if and only if tan(τ) = sin(2ax)e2ya and cos(2ax) = 0. Hence all

x ∈ π

2a(3/2 + 2Z) and y =

1

2aln(tan(τ))

define solutions to the secular equation with non-trivial imaginary part. So, although there is a similarityrelation for the local Cayley transforms, the operator cannot be similar to a self-adjoint one.

28 AMRU HUSSEIN

Example 5.5 (Similarity by geometry). Consider the interval [0, a] and impose the boundary condition

ψ′(0) + (iα− β)ψ(0) and ψ′(a) + (iα+ β)ψ(a) = 0, for α, β ∈ R,

cf. [KBZ06, Sec. 6.3] which in matrix notation becomes

A =

[

iα− β 00 −(iα+ β)

]

and B =

[

1 00 1

]

.

In [KBZ06] it has been shown that β = 0 the spectrum is real, and if α 6= nπ/a, n ∈ N, then −∆(A,B) issimilar to a self-adjoint operator. This highlights that similarity of Laplacians on graphs can be achievedalso without similarity of the local boundary conditions.

These boundary conditions were introduced in [KBZ06] and studied further in [Kre08]. The casewith β 6= 0 is studied in [KS10], and the general case of Robin boundary conditions on each endpoint,including a discussions of the previously mentioned cases, is studied in [KSZ14] where some similar-ity transforms are computed explicitly, see also [KKNS12, HCKS11, KLZ18] for an extension of thismodel. A generalisation of this example to metric graphs was proposed in [Zno15]. The eigenvalues andeigenvectors have already been studied in [Mih62] and [DS71, Sec.XIX.3].

5.2. Proof of Theorem 5.1. For the proof of Theorem 5.1, it is essential that for star graphs ageneralized reflection principle applies. Odd and even refection along a hyperplane is often applied topass from second order elliptic differential operators on the whole space to half space problems withDirichlet and Neumann boundary conditions, respectively. The generalized reflection principle for stargraphs proposed in [HKS15, Section 6] gives that similarity of the boundary conditions implies similarityof the Laplace operators. More precisely, if I = ∅, then for

A = G−1A′G and B = G−1B′G

one can consider the map

ΦG : Dom(∆(A,B)) → Dom(∆(A′, B′)), ΦG{ψi}i∈E = {GIE,Eψi}i∈E

which induces the similarity

∆(A,B) = ΦG−1∆(A′, B′)ΦG.

Note that if −∆(A,B) is similar to a self-adjoint operator, i.e., in particular a normal one, thenby Corollary 4.14, the boundary conditions are quasi-sectorial. Using the similarity transform ΦG, see[HKS15, Section 6], one can consider without loss of generality boundary conditions of the form 3.2.Then only the following cases can occur

(a) σ(L) ∩CRe>0 \ (0,∞) 6= ∅,(b) σ(L) ∩CRe>0 \ (0,∞) = ∅, σ(L) ∩ iR \ {0} 6= ∅ or l ∈ σ(L) ∩ [0,∞) 6= ∅ and there is a cyclic vector

to the eigenvalue l;(c) σ(L) ⊂ CRe<0 ∪ [0,∞), where l ∈ σ(L) ∩ [0,∞) has no cyclic vector.

The key observation of the proof of Theorem 5.1 is that poles of the Cayley transform are also poles ofthe resolvent, or of some meromorphic extension of the resolvent to the unphysical sheet.

For case (a), note that k2 ∈ σ(−∆(A,B)) if for Im k > 0 det(A + ikB) = 0. For quasi-sectorialboundary conditions this means that det(L+ ik) = 0, and if l ∈ σ(L) ∩CRe>0 \ (0,∞), then k = il is azero of det(L + ik) = 0 with Im k > 0, and hence −l2 ∈ C \ R is an eigenvalue of −∆(A,B). Since thespectrum is stable under similarity transforms, −∆(A,B) cannot be similar to a self-adjoint operator.

For case (b), recall that for I = ∅ one has σr(−∆(A,B)) = ∅, see [HKS15, Proposition 4.6], andσp(−∆(A,B)) ⊂ C \ [0,∞) consists of finitely many eigenvalues with finite multiplicity. Moreover,σess(−∆(A,B)) = [0,∞), and if this operator is similar to a self-adjoint one, then there exists anequivalent metric ‖·‖′ in L2(G) such that for some C > 0

C‖(−∆(A,B)− λ± iε)−1‖ ≤ ‖(−∆(A,B)− λ± iε)−1‖′ = ε−1 for ε > 0, λ ∈ σ(−∆(A,B)) ⊂ R.

This would imply that for any −∆(A′, B′) which is self-adjoint with respect to the standard scalarproduct

ε‖(−∆(A,B)− λ± iε)−1 − (−∆(A′, B′)− λ± iε)−1‖ ≤ 1/C + 1 for ε > 0, λ ∈ R.(5.2)


In case (b) by Lemma 3.6 the resolvent has some singularities leading to a contractions to (5.2).First, if l ∈ σ(L)∩ i(−∞, 0), then det(A+ ikB) has a zero at k = −l ∈ (0,∞), and hence S(k,A,B)

has a pole at −l > 0. If l ∈ σ(L) ∩ i(0,∞), then l ∈ σ(L∗) ∩ i(−∞, 0), and hence the Cayley transformfor the adjoint operator S(k,A′, B′) with A′ = L∗+P and B′ = P⊥ has a pole −l > 0. Since similarityof −∆(A,B) and −∆(A,B)∗ = −∆(A′, B′) to a self-adjoint operator is equivalent, it is sufficient toconsider one of the two operators, and one can assume without loss of generality that l ∈ σ(L)∩i(−∞, 0).

Let v be an eigenvector to l, then consider the normalized function ψ defined by ψ(x) := (1/a)χ[0,a]v,and for Im k > 0

ϕk : = (−∆(A,B)− k2)−1ψ − (−∆(0,1)− k2)−1ψ

=i

2kφE (x, k)(S(k,A,B) − 1)

(ˆ a

0

eikydy

)

v

=i

2kφE (x, k)(−

l − ik

l + ik− 1)

(ˆ a

0

eikydy

)

v.

For k = Re k + i Imk one obtains

λk := Re k2 = (Re k)2 − (Im k)2 and εk := Im k2 = 2(Re k) · (Im k).

Let l = −iξ, ξ > 0, one fixes Re k = ξ, and then for Im k > 0 using that ‖φE(·, k)v‖ = ‖v‖/√2 Im k for

v ∈ Cd

εk‖ϕk‖ = 2ξ · (Im k)1

2|k|1√

2 Imk

2ξ

| Im k||eika − 1|

|k| ‖v‖

=|eika − 1|

|k|22ξ2√2 Im k

‖v‖

≥ cξ2

ξ2 + (Im k)21√Im k

→ ∞ as Im k → 0,

where one uses that for k = Re k + i Im k with Re k, Im k ∈ R and Re k 6= 0 there exists a c > 0 suchthat |eika − 1| > c as Im k → 0. This is a contradiction to (5.2). Note that the singular term 1/

√Im k

is due to a singularity of the Cayley transform.Consider now the case l ∈ σ(L) ∩ [0,∞) where v with ‖v‖ = 1 is a cyclic vector to the eigenvalue l

with Lv = (l + N)v for a nilpotent matrix N with Nv 6= 0 and N2 = 0. From (3.6) and (3.7) in theproof of Lemma 3.6 one deduces that then

S(k,A,B)v =

(

−1+2ik

l + ik

(

1+N

−(l+ ik)

))

v for l+ ik 6= 0.

Then similar to the above

ϕk := (−∆(A,B)− k2)−1ψ − (−∆(1, 0)− k2)−1ψ =i

2kφE(x, k)(S(k,A,B) + 1)

(ˆ a

0

eikydy

)

v

=i

2kφE(x, k)

2ik

l + ik

(

1+N

−(l+ ik)

)(ˆ a

0

eikydy

)

v.

For l = 0, assume that Re k = Im k, i.e., λk = 0, εk = 2(Im k)2, and |k2| = 2(Im k)2. Then

εk‖ϕk‖ = 2(Im k)21

2|k|1√

2 Imk

2

|k| ‖Nv‖|eika − 1|

|k|

=|eika − 1|

|k|1√

2 Im k‖Nv‖ → ∞ as Im k → 0,

where |eika−1||k| → a as k → 0. As above this is a contradiction to (5.2).

30 AMRU HUSSEIN

For l > 0, assume that Im k = l, i.e., λk = (Re k)2 − l2, εk = 2l(Re k). Then

εk‖ϕk‖ = 2l(Re k)1

2|k|1√2l

2|k||l + ik|

1

|l+ ik| ‖−(l+ ik)v +Nv‖ |eika − 1||k|

=1√2l

1

(Re k)‖−(l+ ik)v +Nv‖ |e

ika − 1||k|

≈ 1

(Re k)

1√2l‖Nv‖ |e

−la − 1||l| → ∞ as Re k → 0,

where the singular term 1/(Re k) again comes from a singularity of the Cayley transform, thus contra-dicting (5.2).

Now, consider case (c). First, note that the spectrum is real, and it consists of σess(−∆(A,B)) =[0,∞) plus an at most finite set of negative eigenvalues with finite multiplicity. Characterizations forthe similarity of operators with real spectrum to self-adjoint operators have been proven more or lesssimultaneously in different variants by van Casteren [vC83, Theorem 3.1], Naboko [Nab84, Theorem 2],and Malamud [Mal85, Theorem 1].

Theorem 5.6 (cf. Theorem 2 in [Nab84] and Theorem 1 [Mal85]). A closed densely defined linearoperator T in a Hilbert space H is similar to a self-adjoint operator if and only if its spectrum is realand there exists a constant C > 0 such that for all g ∈ H

supε>0

ˆ

R+iε

‖(T − z)−1g‖2dz ≤ C‖g‖2,

supε>0

ˆ

R+iε

‖(T ∗ − z)−1g‖2dz ≤ C‖g‖2.

This theorem has been applied in a similar situation in [GK14]. Here, the estimate in Theorem 5.6could be verified directly since the resolvent kernel is given explicitly, but it involves a rather lengthycomputation which can be avoided by the series of comparison arguments given above.

In the situation of (c), in the Jordan normal form for the matrix L from the quasi-Weierstrass normalform the eigenvalues in [0,∞) correspond to a diagonal part LReL≥0 while the eigenvalues in CRe<0

correspond to a possibly non-diagonal block LRe<0. Hence, the matrices A,B are similar to

A′ =

LReL≥0 0 00 LRe<0 00 0 1

and B′ =

1 0 00 1 00 0 0

with σ(LReL≥0) ⊂ (0,∞) and LReL≥0 diagonal, and σ(LRe<0) ⊂ CRe<0. Due to the generalizedreflection principle on star graphs, −∆(A,B) is similar to −∆(A′, B′), cf. [HKS15, Theorem 6.2] Nowdefine

Asa =

LReL≥0 0 00 0 00 0 1

and Bsa =

1 0 00 1 00 0 0

.

Then ∆(Asa, Bsa) is a self-adjoint operator, and

S(k,A′, B′)−S(k,Asa, Bsa) =

0 0 00 S(k, LRe<0,1) 00 0 0

.

The poles of S(k, LRe<0,1) ly in the lower half plane with CIm<0, compare Lemma 3.6, and sinceLRe<0,1 define quasi-sectorial boundary conditions one has by Lemma 3.7 that S(k, LRe<0,1) is uni-formly bounded on CIm>0. Hence, for some C > 0

(5.3) ‖(−∆(A′, B′)− k2)−1ψ − (−∆(Asa, Bsa)− k2)−1ψ‖2

= ‖ i

2kφE(·, k) (S(k,A′, B′)−S(k,Asa, Bsa))

ˆ

GφE(y, k)ψ(y)dy‖2 ≤ C

1

4|k|2d

2 Im k‖ˆ

GφE (y, k)ψ(y)dy‖2


for any ψ ∈ L2(G) since ‖φE(·, k)‖2 = d/(2 Imk) for Im k > 0. Next note that since S(k,1, 0) = −1and S(k, 0,1) = 1

‖(−∆(1, 0)− k2)−1ψ − (−∆(0,1)− k2)−1ψ‖2 = ‖ ikφE(x, k)

ˆ ∞

0

φE(y, k)ψ(yj)dy‖2

=1

|k|21

2 Imk‖ˆ ∞

0

φE (y, k)ψ(yj)dy‖2,

and since both operators −∆(1, 0) and ∆(0,1) are self-adjoint, by Theorem 5.6 for Im√· > 0 there

exists another constant C > 0 such that

supε>0

ˆ

R+iε

1

2|z| · Im√z‖ˆ

GφE(y,

√z)ψ(y)dy‖2dz ≤ C‖ψ‖2 for all ψ ∈ L2(G).

Consequently, by (5.3) there exists a possibly larger constant C > 0 such that for all ψ ∈ L2(G)

supε>0

ˆ

R+iε

‖(−∆(A′, B′)− z)−1ψ‖2dz ≤ supε>0

ˆ

R+iε

‖(−∆(A′, B′)− z)−1ψ− (−∆(Asa, Bsa)− z)−1ψ‖2dz

+ supε>0

ˆ

R+iε

‖(−∆(A′, B′)− z)−1ψ‖2dz ≤ C‖ψ‖2.

For (−∆(A,B)∗−z)−1 an analogous argument applies, where one uses that the by [HKS15, Proposition3.7] boundary conditions for the adjoint operator −∆(Aad, Bad) = −∆(A,B)∗ are given by

Aad := −1

2(S(k,A,B)∗ − 1) and Bad :=

1

−2ik(S(k,A,B)∗ + 1) .

Hence

S(−k,Aad, Bad) = S(k,A,B)∗,

and its poles behave as the ones of the adjoint since for Aad, Bad the matrix Lad in the quasi-Weierstrassnormal form can be chosen as Lad = L∗ with L determined by A,B. Hence by Theorem 5.6 −∆(A′, B′)is similar to a self-adjoint operator and by the generalized reflection principle used before also −∆(A,B)is similar to a self-adjoint operator. �

Acknowledgement. I would like to thank David Krejcirık, Delio Mugnolo, and Petr Siegl for manyhelpful discussions and insights on the subtleties of non-self-adjointness in operator theory.

References

[ABHN01] W. Arendt, C. J. K. Batty, M. Hieber, and F. Neubrander, Vector-Valued Laplace Transforms and Cauchy

Problems, Monographs in Mathematics, vol. 96, Birkhauser, Basel, 2001.[AFK02] S. Albeverio, S.-M. Fei, and P. Kurasov, Point interactions: PT -Hermiticity and reality of the spectrum, Lett.

Math. Phys. 59 (2002), no. 3, 227–242. MR1904984[Agr94] M. S. Agranovich, On series in root vectors of operators defined by forms with a selfadjoint principal part,

Funktsional. Anal. i Prilozhen. 28 (1994), no. 3, 1–21, 95. MR1308388[AK05] S. Albeverio and S. Kuzhel, One-dimensional Schrodinger operators with P-symmetric zero-range potentials,

J. Phys. A 38 (2005), no. 22, 4975–4988. MR2148637[AKU15] M. Astudillo, P. Kurasov, and M. Usman, RT -symmetric Laplace operators on star graphs: real spectrum and

self-adjointness, Adv. Math. Phys. (2015), Art. ID 649795, 9.[AM84] F. Ali Mehmeti, Problemes de transmission pour des equations des ondes lineaires et quasilineaires, Hyperbolic

and holomorphic partial differential equations, 1984, pp. 75–96. MR747657[AM94] F. Ali Mehmeti, Nonlinear waves in networks, Mathematical Research, vol. 80, Akademie-Verlag, Berlin, 1994.

MR1287844[BC13] A. Bobrowski and W. Chojnacki, Cosine families and semigroups really differ, J. Evol. Equ. 13 (2013), no. 4,

897–916. MR3127029[BE09] J. Bolte and S. Endres, The trace formula for quantum graphs with general self adjoint boundary conditions,

Ann. Henri Poincare 10 (2009), no. 1, 189–223. MR2496307[BEH08] J. Blank, P. Exner, and M. Havlıcek, Hilbert space operators in quantum physics, Second, Theoretical and

Mathematical Physics, Springer, New York; AIP Press, New York, 2008. MR2458485[Ben07] C. M. Bender, Making sense of non-Hermitian Hamiltonians, Rep. Progr. Phys. 70 (2007), no. 6, 947–1018.

MR2331294

32 AMRU HUSSEIN

[BFN16] J. Banasiak, A. Falkiewicz, and P. Namayanja, Semigroup approach to diffusion and transport problems on

networks, Semigroup Forum 93 (2016), no. 3, 427–443. MR3572410[Bir08a] G. D. Birkhoff, Boundary value and expansion problems of ordinary linear differential equations, Trans. Amer.

Math. Soc. 9 (1908), no. 4, 373–395. MR1500818[Bir08b] , On the asymptotic character of the solutions of certain linear differential equations containing a

parameter, Trans. Amer. Math. Soc. 9 (1908), no. 2, 219–231. MR1500810[BIT12] T. Berger, A. Ilchmann, and S. Trenn, The quasi-Weierstraßform for regular matrix pencils, Linear Algebra

Appl. 436 (2012), no. 10, 4052–4069. MR2914563[BK13] G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, Mathematical Surveys and Monographs,

vol. 186, American Mathematical Society, Providence, RI, 2013. MR3013208[Bob12] A. Bobrowski, From diffusions on graphs to Markov chains via asymptotic state lumping, Ann. Henri Poincare

13 (2012), no. 6, 1501–1510. MR2966471[DS71] N. Dunford and J. T. Schwartz, Linear operators. Part III: Spectral operators, Interscience Publishers [John

Wiley & Sons, Inc.], New York-London-Sydney, 1971. MR0412888[DZ06] R. Dager and E. Zuazua, Wave propagation, observation and control in 1-d flexible multi-structures,

Mathematiques & Applications (Berlin) [Mathematics & Applications], vol. 50, Springer-Verlag, Berlin, 2006.MR2169126

[EKF19] K.-J. Engel and M. Kramar Fijavz, Waves and diffusion on metric graphs with general vertex conditions, Evol.Equ. Control Theory 8 (2019), no. 3, 633–661. MR3985968

[Fre12] G. Freiling, Irregular boundary value problems revisited, Results Math. 62 (2012), no. 3-4, 265–294. MR2990164[GK14] A. Grod and S. Kuzhel, Schrodinger operators with non-symmetric zero-range potentials, Methods Funct.

Anal. Topology 20 (2014), no. 1, 34–49. MR3242121

[GK69] I. C. Gohberg and M. G. Kreın, Introduction to the theory of linear nonselfadjoint operators, Translatedfrom the Russian by A. Feinstein. Translations of Mathematical Monographs, Vol. 18, American MathematicalSociety, Providence, R.I., 1969. MR0246142

[GM20] F. Gregorio and D. Mugnolo, Bi-Laplacians on graphs and networks, J. Evol. Equ. 20 (2020), no. 1, 191–232.MR4072654

[Har00] M. Harmer, Hermitian symplectic geometry and extension theory, J. Phys. A 33 (2000), no. 50, 9193–9203.MR1804888

[HCKS11] H. Hernandez-Coronado, D. Krejcirık, and P. Siegl, Perfect transmission scattering as a PT -symmetric spectral

problem, Phys. Lett. A 375 (2011), no. 22, 2149–2152. MR2799703[HKS15] A. Hussein, D. Krejcirık, and P. Siegl, Non-self-adjoint graphs, Trans. Amer. Math. Soc. 367 (2015), no. 4,

2921–2957. MR3301887[HM20] A. Hussein and D. Mugnolo, Laplacians with point interactions – expected and unexpected spectral properties,

Semigroups of operators theory and applications, 2020.[Hus13] A. Hussein, Bounds on the negative eigenvalues of Laplacians on finite metric graphs, Integral Equations

Operator Theory 76 (2013), no. 3, 381–401. MR3065300[Hus14] , Maximal quasi-accretive Laplacians on finite metric graphs, J. Evol. Equ. 14 (2014), no. 2, 477–497.

MR3207624[JMZ15] B. Jacob, K. Morris, and H. Zwart, C0-semigroups for hyperbolic partial differential equations on a one-

dimensional spatial domain, J. Evol. Equ. 15 (2015), no. 2, 493–502. MR3353144[Kat95] T. Kato, Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995.

Reprint of the 1980 edition. MR1335452[KBZ06] D. Krejcirık, H. Bıla, and M. Znojil, Closed formula for the metric in the Hilbert space of a PT -symmetric

model, J. Phys. A 39 (2006), no. 32, 10143–10153. MR2252706[KF00] A. V. Kiselev and M. M. Faddeev, The similarity problem for non-self-adjoint operators with absolutely con-

tinuous spectrum, Funct Anal Its Appl 34 (2000), 143–145.[KFMS07] M. Kramar Fijavz, D. Mugnolo, and E. Sikolya, Variational and semigroup methods for waves and diffusion

in networks, Appl. Math. Optim. 55 (2007), no. 2, 219–240. MR2305092[Kis09] A. V. Kiselev, Similarity problem for non-self-adjoint extensions of symmetric operators, Methods of spectral

analysis in mathematical physics, 2009, pp. 267–283. MR2732083[Kis11] , The similarity problem for non-selfadjoint operators with absolutely continuous spectrum: restrictions

to spectral subspaces, Spectral theory and analysis, 2011, pp. 67–82. MR2808464[KKNS12] D. Kochan, D. Krejcirık, R. Novak, and P. Siegl, The Pauli equation with complex boundary conditions, J.

Phys. A 45 (2012), no. 44, 444019, 14. MR2991886[Klo12] B. Kloss, The flow approach for waves in networks, Oper. Matrices 6 (2012), no. 1, 107–128. MR2952439[KLZ18] D. Krejcirık, V. Lotoreichik, and M. Znojil, The minimally anisotropic metric operator in quasi-Hermitian

quantum mechanics, Proc. A. 474 (2018), no. 2217, 20180264, 13. MR3875213[KPS07] V. Kostrykin, J. Potthoff, and R. Schrader, Heat kernels on metric graphs and a trace formula, Adventures in

mathematical physics, 2007, pp. 175–198. MR2423580[KPS08] , Contraction semigroups on metric graphs, Analysis on graphs and its applications, 2008, pp. 423–458.

MR2459885[KPS12a] , Brownian motions on metric graphs, J. Math. Phys. 53 (2012), no. 9, 095206, 36. MR2905788


[KPS12b] , Finite propagation speed for solutions of the wave equation on metric graphs, J. Funct. Anal. 263(2012), no. 5, 1198–1223. MR2943728

[Kre08] D. Krejcirık, Calculation of the metric in the Hilbert space of a PT -symmetric model via the spectral theorem,J. Phys. A 41 (2008), no. 24, 244012, 6. MR2455810

[Kre19] , Complex magnetic fields: an improved Hardy-Laptev-Weidl inequality and quasi-self-adjointness,SIAM J. Math. Anal. 51 (2019), no. 2, 790–807. MR3924615

[KS06] V. Kostrykin and R. Schrader, Laplacians on metric graphs: eigenvalues, resolvents and semigroups, Quantumgraphs and their applications, 2006, pp. 201–225. MR2277618

[KS10] D. Krejcirık and P. Siegl, PT -symmetric models in curved manifolds, J. Phys. A 43 (2010), no. 48, 485204,30. MR2738140

[KS99] V. Kostrykin and R. Schrader, Kirchhoff’s rule for quantum wires, J. Phys. A 32 (1999), no. 4, 595–630.MR1671833

[KSTV15] D. Krejcirık, P. Siegl, M. Tater, and J. Viola, Pseudospectra in non-Hermitian quantum mechanics, J. Math.Phys. 56 (2015), no. 10, 103513, 32. MR3416732

[KSZ14] D. Krejcirık, P. Siegl, and J. Zelezny, On the similarity of Sturm-Liouville operators with non-Hermitian

boundary conditions to self-adjoint and normal operators, Complex Anal. Oper. Theory 8 (2014), no. 1, 255–281. MR3147722

[Kuc02] P. Kuchment, Graph models for waves in thin structures, Waves Random Media 12 (2002), no. 4, R1–R24.MR1937279

[Kuc04] , Quantum graphs. I. Some basic structures, 2004, pp. S107–S128. Special section on quantum graphs.MR2042548

[KZ17] S. Kuzhel and M. Znojil, Non-self-adjoint Schrodinger operators with nonlocal one-point interactions, BanachJ. Math. Anal. 11 (2017), no. 4, 923–944. MR3708536

[Lan75] H. J. Landau, On Szego’s eigenvalue distribution theorem and non-Hermitian kernels, J. Analyse Math. 28(1975), 335–357. MR487600

[LLS94] J. E. Lagnese, G. Leugering, and E. J. P. G. Schmidt, Modeling, analysis and control of dynamic elastic multi-

link structures, Systems & Control: Foundations & Applications, Birkhauser Boston, Inc., Boston, MA, 1994.MR1279380

[Loc00] J. Locker, Spectral theory of non-self-adjoint two-point differential operators, Mathematical Surveys and Mono-graphs, vol. 73, American Mathematical Society, Providence, RI, 2000. MR1721499

[Mal85] M. M. Malamud, A criterion for a closed operator to be similar to a selfadjoint operator, Ukrain. Mat. Zh. 37(1985), no. 1, 49–56, 134. MR780914

[Man10] S. S. Man’ko, On δ′-like potential scattering on star graphs, J. Phys. A 43 (2010), no. 44, 445304, 14.MR2733833

[Man15] S. S. Manko, On δ′-couplings at graph vertices, Mathematical results in quantum mechanics, 2015, pp. 305–313.MR3381162

[Mih62] V. P. Mihaılov, On Riesz bases in L2(0, 1), Dokl. Akad. Nauk SSSR 144 (1962), 981–984. MR0149334[MM03] R. Mennicken and M. Moller, Non-self-adjoint boundary eigenvalue problems, North-Holland Mathematics

Studies, vol. 192, North-Holland Publishing Co., Amsterdam, 2003. MR1995773[Mos06] A. Mostafazadeh, Delta-function potential with a complex coupling, J. Phys. A 39 (2006), no. 43, 13495–13506.

MR2269700[Mos10] , Pseudo-Hermitian representation of quantum mechanics, Int. J. Geom. Methods Mod. Phys. 7 (2010),

no. 7, 1191–1306. MR2749385[Mug07] D. Mugnolo, Gaussian estimates for a heat equation on a network, Netw. Heterog. Media 2 (2007), no. 1,

55–79. MR2291812[Mug14] , Semigroup methods for evolution equations on networks, Understanding Complex Systems, Springer,

Cham, 2014. MR3243602[Mug19] , What is actually a metric graph?, 2019.

[Nab84] S. N. Naboko, Conditions for similarity to unitary and selfadjoint operators, Funktsional. Anal. i Prilozhen.18 (1984), no. 1, 16–27. MR739086

[Nai67] M. A. Naimark, Linear differential operators. Part I: Elementary theory of linear differential operators, Fred-erick Ungar Publishing Co., New York, 1967. MR0216050

[Pav79] B. S. Pavlov, The basis property of a system of exponentials and the condition of Muckenhoupt, Dokl. Akad.Nauk SSSR 247 (1979), no. 1, 37–40. MR545940

[SGH92] F. G. Scholtz, H. B. Geyer, and F. J. W. Hahne, Quasi-Hermitian operators in quantum mechanics and the

variational principle, Ann. Physics 213 (1992), no. 1, 74–101. MR1144600[Sie09] P. Siegl, Surprising spectra of PT-symmetric point interactions, 2009.[Sie11] , Systeme quantitique non-Hermitiens, representations indecomposables et quantification par etats

coherents, Ph.D. Thesis, 2011.[vB88] J. von Below, Classical solvability of linear parabolic equations on networks, J. Differential Equations 72

(1988), no. 2, 316–337. MR932369[vC83] J. A. van Casteren, Operators similar to unitary or selfadjoint ones, Pacific J. Math. 104 (1983), no. 1, 241–

255. MR683741

34 AMRU HUSSEIN

[Zno15] M. Znojil, Quantum star-graph analogues of pt -symmetric square wells: Part ii, spectra, Canadian Journal ofPhysics 93 (2015), no. 7, 765–768, available at https://doi.org/10.1139/cjp-2014-0475 .

Department of Mathematics, TU Kaiserslautern, Paul-Ehrlich-Straße 31, 67663 Kaiserslautern, Germany

E-mail address: [email protected]

https://doi.org/10.1139/cjp-2014-0475

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